Properties of Groups Acting on Twin-Trees and Chabauty Space

PROPERTIES OF GROUPS ACTING ON TWIN-TREES AND CHABAUTY SPACE

Robert Kelvey

A Dissertation

Submitted to the Graduate College of Bowling Green State University in partial fulﬁllment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

December 2016

Committee:

Rieuwert Blok, Advisor

Lee Nickoson, Graduate Faculty Representative

Mihai Staic

Xiangdong Xie ii ABSTRACT

Rieuwert Blok, Advisor

In this dissertation, we study groups that act on twin trees. A twin tree consists of a pair of (inﬁnite) simplicial trees (푇 +, 푇 −) that are “twinned” by means of a co-distance function 훿*, which assigns a non-negative integer to pairs of vertices from each tree. If 푛 = 훿*(푥+, 푦−) for vertices 푥+ in 푇 + and 푦− in 푇 −, then we think of 푥+ and 푦− as having distance ∞ − 푛. An example of a

twin tree is T = (T +, T −, 훿*), where T + and T − are the associated Bruhat-Tits trees arising from two different discrete valuations on the ﬁeld 헄(푡). A group 퐺 acts on a twin tree 푇 = (푇 +, 푇 −, 훿*) if it acts on each tree 푇 ± and preserves

the co-distance function. For the twin tree T arising from discrete valuations on 헄(푡), the group

−1 GL2(헄[푡, 푡 ]) naturally acts on the twinning. The subgroup GL2(헄[푡]) stabilizes a vertex of the the

+ − tree T . The action of GL2(헄[푡]) on T yields a fundamental domain an inﬁnite ray, and from this action one obtains Nagao’s Theorem:

∼ GL2(헄[푡]) = GL2(푘) *퐵(헄) 퐵(헄[푡]),

where note 퐵(헄) and 퐵(헄[푡]) are subgroups of upper triangular matrices in GL2(헄) and GL2(헄[푡]),

−1 respectively. In this work, we investigate the fundamental domains for subgroups 퐺 < GL2(헄[푡, 푡 ])

that stabilize subtrees of the tree T +. For a general group 퐺 acting on a twin-tree, we consider its space of closed subgroups 풞(퐺), called the Chabauty space. By constructing a left-invariant metric on the underlying automorphism group of the twin-tree, one can endow 풞(퐺) with a metric as well. Using this, we study the distance between vertex stabilizer subgroups in 퐺. This will hopeful lead to future work generalizing the

−1 special case of T and GL2(헄[푡, 푡 ]). iii ACKNOWLEDGMENTS

There are many, many people for whom, without their direct impact on my life, I would not be where I am today. Certainly I would not be writing an acknowledgements section to a dissertation! So I am going to take this opportunity to get personal. I’m going to start somewhat near the beginning, so my apologies but - this might take while. Firstly, I would like to thank Mr. Klima, my AP Calculus teacher in High School. My current opinion on AP Calculus withstanding, I truly would not have even thought to continue with mathematics in College were it not for Mr. Klima. At the time, I assumed I would major in computer science. I knew that would require some more mathematics, but had no idea of how much or really of what kind (all I knew was that I was scared of “proving things”). Mr Klima pointed out that, most likely one would be close to a minor in math by the end of a computer science degree, so why not just go ahead and do a double major? Until that moment, such a thought had never entered my mind. But as soon as I was made aware of such a possibility, the thought stayed with me. That is what I wanted to do. So thank you Mr. Klima, for helping to get me started. Secondly, I would like to thank all of my teachers at Carroll Community College. I know now that all of them were teachers because they love teaching. They love the students and the subject. Perhaps it is from all of you that I was instilled with my first desires to teach. In particular, I would like to thank Rob Brown (a fellow McDaniel alum!) for giving me the chance to take a math class my first semester, despite there being no actual class to take. And of course, Dr. Raza! Although you may, on occasion, say some outlandish things, your enthusiasm for Chemistry and obvious care for students had a great impact on me. I must also thank all those with whom I worked with at the Academic Center at Carroll Community College, especially for giving me the opportunity to tutor students, even after I left and was a student at McDaniel College. The tutoring experiences I gained at Carroll gave me the foundation I needed to become a teacher. Thank you all! None of what follows would be at all possible were it not for the amazing Mathematics and Computer Science department at McDaniel College. You challenged and pushed me academically, further than I thought I could go. I certainly would have never thought to take Topology before iv having Abstract Algebra or Analysis without your motivations and encouragement; nor would I have taken Cryptography that first Spring semester. It was certainly difficult (at times, very difficult). So thank you Sara, Spencer, Italo, Pavel, and (may you rest in peace) Harry. You were the giants that helped me first touch the sky. So much has happened in the last six years at Bowling Green State University, and there are many people to thank. I will be forever grateful that I got to spend my first year as a graduate assistant in the Learning Commons, since who knows if I would even be writing this without that first opportunity. Almost every faculty and staff member at the BGSU Math and Stats department deserves thanks for assisting me, even if it was in some minute way on a random day. So if I don’t mention you by name below, my apologies! But thank you all. I would like to thank Dr. Neal Carothers for letting me share in his last year of teaching at BGSU. You taught me to stop worrying and enjoy Analysis. And thank you to Dr. Steven Seubert for powering me through my comprehensive exams, and for teaching me to be confident and know when my words are true. And thank you to Dr. So-Hsiang Chou for teaching me mystical things, along with some applied math (which really isn’t so bad). I will forever emulate part of your teaching style as well! Thank you to Dr. Chan, Dr. Meel, and Dr. Chen for your service as department chair during my time at BGSU. And thank you to Dr. John Chen, Dr. Sun, Dr. Blok, and Dr. Zirbel for your time and efforts as Graduate Coordinators. Your work, effort and advice has certainly helped me considerably over the years. And I absolutely would not be writing this without the financial support of the department. Special thanks are certainly in order for the office staff: Mary, Marcia, and Barb, thank you all so much for the work that you do/did (congratulations on retirement!). I certainly must thank all of the Algebraists, as each of you has helped me in your own way. Thank you Dr. Xiangdong Xie for all your teaching, advising, and invitations. Dr. Mihai Staic deserves an MVP award for assisting me my last year of school. Your assistance with my job application process cannot be understated in its helpfulness. Thank you!! And of course, thank you Dr. Rieuwert Blok. Even across the pond, you found time to aid me through my dissertation, v job search, and life itself. Your humor and impressive ability to write quickly on the board will forever stay with me. Thank you for being my advisor and helping me get this document to its final destination. I have had many classmates and made many friends over the years. Even if I haven’t spoken to you in a long time, I would like to thank you for being my friend, even if just for a short time. My fellow McDaniel crew: John, Ben, Stephen, Feng, Shannon, Wesley, and David (both of you) - thank you; I’ll hopefully see you at a homecoming someday in the future! My first friends in Bowling Green were those of the Men’s and Women’s BGSU Cross Country team. All my teammates, I am proud to have run an incredible season with all of you. Thank you Ben for giving me a call and making me feel so welcome. Maybe I’ll scramble a mountain with you or we can run an ultra someday. Thank you Megan and Tara for M3 at El Zarape! Jesse and Aaron Smuda, you are two cool brothers. And of course, John Bernard: you are amazing and a wonderful friend. I am glad to have met you all! Thank you to all my fellow Math and Stat graduate students. We have laughed, cried, and drank at Stones Throw together many a good time. You all helped me keep my sanity. Special thanks to Jen, Rachelle, Chas, and Leo for helping me survive my first two years. Thank you to all past and future 470A-ers, especially Leo, Gokul, Chas, John, Nathan, Mark, and Jake. Having classmates, officemates and friends like you has been a blessing. Thank you! And best of luck to those of you still making your way through. I’ll see you at future JMMs! And a super special heartfelt thank you to Colleen. I am so very happy you walked over that day from the ASOR picnic to say hi to the Math nerds. Little did you know you’d walk away from there with my heart forever. Thank you!! Last, but certainly not least, I would like to thank my family. Thank you to my brothers for their love and support (in their own brotherly ways). Thank you Pop-pop, for helping pay for that last little bit of McDaniel tuition. You’re a shining star. Thank you to my father William, for helping me get (and stay) on my feet for college and graduate school. You are an awesome, unflappable dad. Thank you to my mother Yvonne, for your nurturing love, friendship, and unwavering support. You literally rock and you know it! vi Alright. I think that about does it for acknowledgments. Even if you were not mentioned by name above, please know that I indeed thank you and love you from the bottom of my heart. Now let’s do some mathematics, shall we?

Bowling Green, August 2016 Robert Kelvey. vii

TABLE OF CONTENTS

CHAPTER 1 TREES AND TWIN TREES 1 1.1 Graphs and Trees ...... 1 1.2 Quotient Graphs ...... 7 1.3 Bruhat-Tits Tree ...... 14 1.4 Twin Trees ...... 31

CHAPTER 2 GROUPS ACTING ON TREES - BASS-SERRE THEORY 42 2.1 Free Products, Amalgams and HNN Extensions ...... 42 2.2 Groups Acting on Graphs and Bass-Serre Trees ...... 49 2.3 Graphs of Groups ...... 58 2.4 Universal Covering Tree and Quotient Graph of Groups ...... 63 2.5 Nagao’s Theorem ...... 67 2.6 Edge-Indexed Graphs ...... 70 2.7 Properties of Edge-Indexed Graphs ...... 72 2.8 Automorphism Types ...... 78 2.9 Property FA ...... 84 2.10 Nilpotent Groups Acting on a Tree ...... 91

CHAPTER 3 TOPOLOGY OF AUT(X) AND LATTICES 96 3.1 Topological Preliminaries ...... 96 3.2 Topology of Aut(푋) ...... 98 3.3 Other Topologies on Aut(푋) ...... 106 3.4 Lattices in Topological Groups ...... 111 3.5 Tree Lattices ...... 114 3.6 Existence of Tree Lattices ...... 118 viii 3.7 Structure of Tree Lattices ...... 120 3.8 Nagao Lattices ...... 121

CHAPTER 4 FUNDAMENTAL DOMAINS 124

4.1 GL2(퐾) Action on the Bruhat-Tits Tree ...... 124

−1 4.2 The group GL2(헄[푡, 푡 ]) ...... 127

4.3 The action on T − ...... 149 4.4 Future Work ...... 153

CHAPTER 5 CHABAUTY SPACES 155 5.1 Chabauty Topology ...... 155 5.2 Chabauty Topology Metrizability ...... 160 5.3 The Space of Closed Subgroups - Chabauty Space ...... 166 5.4 Metrizability of Topological Groups ...... 174 5.5 The Metric On 퐺 = Aut(푋) ...... 178 5.6 The Chabauty Space for a Twin Tree ...... 187 5.7 Future Work ...... 198

BIBLIOGRAPHY 199 ix PREFACE

In this work we discuss twin trees and groups acting on them. A twin tree consists of a pair of (infinite) simplicial trees (푇 +, 푇 −) that are “twinned” by means of a co-distance function 훿*, which assigns a non-negative integer to pairs of vertices from each tree. If 푛 = 훿*(푥+, 푦−) for vertices 푥+ in 푇 + and 푦− in 푇 −, then we think of 푥+ and 푦− as having distance ∞ − 푛. Hence, vertices of co-distance 0 can be described as “opposites” - being as far apart as possible - a notion that does not exist in either tree on its own. An example of a twin tree T = (T +, T −, 훿*) arises by considering two different Bruhat-Tits trees arising from discrete valuations on the field 헄(푡). In Chapter 1, we give preliminaries on simplicial trees, followed by a precise definition of twin trees with examples. A group 퐺 acts on a twin tree 푇 = (푇 +, 푇 −, 훿*) if it acts on each tree 푇 ± and preserves the co-distance function. For the twin tree T arising from discrete valuations on 헄(푡), the group

−1 GL2(헄[푡, 푡 ]) naturally acts on the twinning. The structure of groups acting on simplicial trees can be given by Bass-Serre theory. The fundamental structure theorem of Bass-Serre theory tells us that a group acting on a tree (without inversions, meaning without swapping adjacent vertices) is necessarily isomorphic to the fundamental group of a graph of groups. Graphs of groups provide a generalization of amalgamated free products and HNN extensions. Conversely, to any graph of groups 픾 there exists a tree on which the fundamental group 휋1(픾) acts. In Chapter 2, we give a thorough account of Bass-Serre theory.

One important example relating to the notion of twin-trees involves the group GL2(헄[푡]). By

studying the action of this group on the twin tree T , one obtains the following theorem of Nagao via Bass-Serre theory: ∼ GL2(헄[푡]) = GL2(푘) *퐵(헄) 퐵(헄[푡]),

where here 퐵(−) stands for the group of upper triangle matrices. More precisely, GL2(헄[푡]) <

−1 + − GL2(헄[푡, 푡 ]) is the stabilizer of a vertex in the tree T . The action of GL2(헄[푡]) on T yields a fundamental domain an inﬁnite ray. From the Bass-Serre theory, one derives the above decom- x position based off this fundamental domain. In Chapter 4 we more closely examine the action of

−1 GL2(헄[푡, 푡 ]) on the corresponding twin tree T to obtain more fundamental domain descriptions. In general, if 푋 is a locally ﬁnite tree, then a topology can be placed on 퐺 = Aut(푋) making 퐺 a locally compact and totally disconnected group. The group 퐺 and its natural action on 푋 is an analog of a semi-simple algebraic group over a local ﬁeld acting on its Bruhat-Tits building, which is itself a generalization of simple Lie groups acting on symmetric spaces. An important class of subgroups in each of these situations is given by lattice subgroups. These are discrete subgroups of

ﬁnite co-volume. In the case of 퐺 = Aut(푋), these are called tree lattices. The group PGL2(헄[푡])

−1 is an example of a tree lattice for 퐺 = PGL2(헄((푡 ))) and its associated Bruhat-Tits tree, which

− is in fact the tree T mentioned above. The lattice PGL2(헄[푡]) is an example of a non-uniform lattice since it has fundamental domain an infinite graph. Moreover, it is a non-uniform lattice of Nagao type (cf. [BL01]). In Chapter 3 we discuss in detail the automorphism group of a locally finite tree 푋 and tree lattices. Theorems of Peter Abramenko (cf. [Abr96]) and Bertrand Re´my (cf. [AR09]) tell us that, for certain groups acting on twin trees, point stabilizers provide non-uniform lattices of Nagao type. This motivated the study of the collection of lattices as a whole and led to considering the space 풞(퐺) of closed subgroups of a group 퐺 acting on a twin tree. Such a space is called the Chabauty space after Claude Chabauty ([Cha50]), who defined a topology on the collection of closed subsets of a topological space, now referred to as the Chabauty topology. This topology is naturally compact regardless of the underlying space. In Chapter 5 we give a detailed discussion of the Chabauty topology and construct a left- invariant metric on 퐺 = Aut(푋) for 푋 a locally finite tree. This allows us to define a metric on the Chabauty space 풞(퐺) and we use this metric to study point stabilizer subgroups of a group acting on a twin tree. 1

CHAPTER 1 TREES AND TWIN TREES

In this chapter, we introduce the notion of graphs, trees, and twin trees. We give a fundamental example of a tree with the construction of the Bruhat-Tits tree arising from a discrete valuation on a ﬁeld. From this, we also describe our main example of a twin tree, involving two Bruhat-Tits trees, arising from separate valuations over the same ﬁeld.

1.1 Graphs and Trees

To talk about trees, we might want to define what a tree is. We do so in a combinatorial fashion, following the definitions first laid out in [Ser80]. See also [Bog08], [DD89], or [Coh89] for these basic definitions.

Remark 1.1.1. One can also think of combinatorial graphs as deﬁned in [BH10], but this is different than the deﬁnition in [Ser80] which we use here.

A tree is a special kind of graph:

Deﬁnition 1.1.2. A graph 푋 is a tuple 푋 = (푉 푋, 퐸푋, 훼, 휔,− ), where 푉 푋 is a nonempty set of vertices, 퐸푋 a set of edges, and mappings 훼 : 퐸푋 → 푉 푋 (beginning), 휔 : 퐸푋 → 푉 푋 (end), and − : 퐸푋 → 퐸푋 (inverse) such that for every 푒 ∈ 퐸푋,

푒 = 푒,

푒 ̸= 푒,

훼(푒) = 휔(푒).

The vertices 훼(푒) and 휔(푒) are called the initial and terminal vertices of the edge 푒. Two vertices 푥 and 푦 are called adjacent if they are the beginning and end of some edge (or vice-versa). If 푥 and 푦 are adjacent, and 푒 is the edge with 훼(푒) = 푥 and 휔(푒) = 푦, then we will sometimes denote the edge 푒 by [푥, 푦]. 2 We often utilize diagrams to describe graphs. Vertices are labeled as points with connecting lines as pairs of inverse edges. Below are some examples.

Example 1.1.3. Here is a simple example of a graph with two vertices connecting by a pair of edges (an edge 푒 with its inverse 푒).

{푒, 푒}

푃 푄

훼(푒) = 푃, 휔(푒) = 푄.

Example 1.1.4. Here is an example of an inﬁnite graph. Think of this graph as the integers: vertices for each 푛 and a pair of inverse edges joining each integer.

풞∞ 푒−1 푒0 푒1 -1 0 1 2

Example 1.1.5. Here is a wilder example of a 3-regular graph: every vertex has 3 edges coming out of it. There are 20 vertices and 30 pairs of inverse edges. 3 If the reader is familiar with some graph theory, then one should note that our deﬁnition gives unoriented graphs. An orientation of a graph 푋 is simply a choice of edge from each pair of inverse edges {푒, 푒} in 푋. The edges picked will be deemed positively oriented, and the set containing

them denoted 푋+. The remaining edges are said to be negatively oriented and are denoted 푋−.

Clearly, if a graph is oriented then 퐸푋 = 푋+ ⊔ 푋−. We say 푋+ is an orientation of 푋. In terms of our diagrams used to convey graphs and trees, we denote an oriented edge with an arrow. The arrowhead will point toward the terminal vertex of the edge. 푒

훼(푒) 휔(푒)

When we discuss the automorphism group of trees and tree lattices, we shall focus on trees that are locally ﬁnite:

Deﬁnition 1.1.6. Let 푋 be a graph. For a vertex 푣 ∈ 푉 푋, the star of 푣, St푋 (푣) = E1(푣), is the set of all edges with initial vertex 푣:

E1(푣) = St푋 (푣) = St(푣) = {푒 ∈ 퐸푋| 훼(푒) = 푣}.

The valency or degree of 푣 is |푆푡(푣)|. We say 푋 is locally ﬁnite if |St(푣)| < ∞ for all 푣 ∈ 퐸푋. We will sometimes write deg(푣) = |St(푣)|, such as in [BL01].

We have a notion of morphism between graphs:

Definition 1.1.7. Let 푋 and 푌 be graphs as defined above. We say 휙 : 푋 → 푌 is a graph morphism or morphism of graphs if 휙 is a map sending vertices to vertices and edges to edges, that satisfies the following conditions:

휙(훼(푒)) = 훼(휙(푒))

휙(휔(푒)) = 휔(휙(푒))

휙(푒) = 휙(푒) 4 Naturally, one can have a graph morphism that is injective, surjective, or bijective. A bijective morphism 휙 : 푋 → 푌 will be called a graph isomorphism. If 휙 : 푋 → 푋 is a graph isomorphism then we say 휙 is a graph automorphism.

푦 푧 푥 휙 휙(푥) 푏 푎 푐 푤 푡 휙(푤) = 휙(푦) 휙(푡) = 휙(푧)

Example 1.1.8. Consider the graph 푋 with vertex set 푉 푋 = {푥, 푦, 푧, 푤, 푡} and the graph 푌 with vertex set 푉 푌 = {푎, 푏, 푐} pictured above; the edge sets are given as indicated in the ﬁgure. We deﬁne a morphism 휙 : 푋 → 푌 by mapping vertices as follows: 휙(푥) = 푎, 휙(푦) = 휙(푤) = 푏, 휙(푧) = 휙(푡) = 푐. The edges are mapped accordingly; for instance, the edge [푥, 푦] in 푋 is mapped to the edge [푎, 푏] in 푌 . Clearly, the morphism 휙 is surjective, but not injective.

Example 1.1.9. Recall the graph 풞∞ Example 1.1.4. The automorphism group for 풞∞ is the inﬁnite

dihedral group 퐷∞ which we now describe.

Any automorphism of 풞∞ is completely determined by where it sends the edge 푒0. Let 푎 and

푏 be automorphisms deﬁned by 푎(푒0) = 푒−1 and 푏(푒0) = 푒1. Then for any integer 푛 ∈ ℤ, we

푛 푛 have 푏 (푒0) = 푒푛 and 푏 푎(푒0) = 푒푛. Hence, we have that the automorphisms 푎 and 푏 generate this

푛 푛 automorphism group, and 퐷∞ = {푏 , 푏 푎 | 푛 ∈ ℤ}.

Remark 1.1.10. Let 휙 : 푋 → 푌 be a morphism of graphs and 푣 ∈ 푉 푋. If 휙|St(푣) has property

푃 (e.g. injective, surjective, etc.), then 휙 is said to be locally 푃 . So for instance, if 휙|St(푣) is an injective map, then we say 휙 is locally injective.

To define what is a tree, we must first define a path in a graph:

Deﬁnition 1.1.11. Let 푋 be a graph. A path 푙 in 푋 is a sequence of edges 푒1, 푒2, . . . , 푒푛 with

휔(푒푖) = 훼(푒푖+1), for 1 ≤ 푖 ≤ 푛 − 1. If 푙 is a path, we will write 푙 = (푒1, 푒2 . . . , 푒푛). The integer

푛 denotes the length of the path, and we will write |푙| = 푛. A path 푙 is closed if 휔(푒푛) = 훼(푒1).

A path 푙 is called reduced if 푒푖+1 ̸= 푒푖, for all 1 ≤ 푖 ≤ 푛 − 1. We will say a path is non-trivial 5 if the length of the path is greater than 0 (a path of length zero is the trivial path that doesn’t go anywhere).

We say 푋 is connected if for any two vertices 푥, 푦 ∈ 푉 푋, there exists a path 푙 = 푒1푒2 . . . 푒푛,

푛 ≥ 1, such that 훼(푒1) = 푥 and 휔(푒푛) = 푦. In this case, we say that 푙 is a path from 푥 to 푦.

푒 푦 2 푧 푒1 푥 푒3

푓1 푓3 푓 푤 2 푡

Example 1.1.12. We may consider several examples of paths on the above graph 푋. The edges in the ﬁgure are oriented as: 푒1 = [푥, 푦], 푒2 = [푦, 푧], 푒3 = [푥, 푧], 푓1 = [푥, 푤], 푓2 = [푤, 푡], 푓3 = [푥, 푡]. The following path traverses the entire graph, traveling clockwise, and beginning and ending at the vertex 푥:

푙 = (푒1, 푒2, 푒3, 푓3, 푓2, 푓1)

Since 푙 begins and ends at 푥, it is a closed path. There are no instances of an edge followed by its inverse, so 푙 is also a reduced path. The following path 푠 is an example of a path that is closed but not reduced:

푠 = (푒1, 푒2, 푒2, 푒2, 푒3, 푓1, 푓1).

Occurrences like 푒2, 푒2 in the path 푠 is called a backtrack or backtracking and the path 푠 is called a path with backtracking. If we remove all instances of backtracking from 푠, then we will have a reduced path:

′ 푠 = (푒1, 푒2, 푒3).

Note that both 푠 and 푠′ are closed paths at 푥.

As illustrated in the above example, if 푙 = (푒1, 푒2, . . . , 푒푖, 푒푖 = 푒푖+1, . . . , 푒푛) is a path with a

′ backtrack from a vertex 푥 to a then the path 푙 = (푒1, 푒2, . . . , 푒푖−1, 푒푖+2, . . . , 푒푛) is also a path from 6 푥 to 푦. So via induction on the length of paths, one can easily see that if there is a path between vertices 푥 and 푦, then there must exist one that is reduced. Now we can deﬁne a tree, using the previous deﬁnitions.

Deﬁnition 1.1.13. A graph 푋 is called a tree if 푋 is connected and contains no non-trivial, closed, reduced paths.

Example 1.1.14. Our ﬁrst examples of graphs above are also examples of trees:

{푒, 푒}

푃 푄

풞∞ 푒−1 푒0 푒1 -1 0 1 2

Example 1.1.15. The graph 푋 below is not a tree, as there exist closed, reduced paths in 푋 that are non-trivial - i.e. there are non-trivial loops in the graph: 푦 푧 푥

푤 푡 푋

To end this section we deﬁne several items concerning trees that will crop up occasionally in the future.

Deﬁnition 1.1.16. In a tree, a reduced path is called a geodesic. One can easily show that, between any two distinct vertices in a tree, there exists a unique geodesic connecting them [Ser80, I.2.2, 7

Prop 8]. Using geodesics, we can deﬁne a graph-theoretic distance function 푑푋 : 푉 푋 × 푉 푋 → ℕ on a tree 푋 by

푑푋 (푥, 푦) = 푛 where 푛 is the length of the unique geodesic connecting 푥 and 푦. A maximal subtree in a connected graph 푋 is a subgraph that is a tree, and has 푉 푇 = 푉 푋. Note that via Zorn’s lemma, maximal subtrees always exist. We simply order subtrees by inclusion, and then can show that maximal with respect to inclusion implies the condition on vertices. Trees are bipartite graphs, meaning if 푋 is a tree, then we can always partition the set of vertices into two classes, 푉 푋 = 푋0 ⊔ 푋1, called the type of the vertex. Speciﬁcally, we declare two vertices 푥 and 푦 in tree 푋 to be of the same type if 푑푋 (푥, 푦) ≡ 0 (mod 2). Put another way, we can think of the type via a function

푑푋 휏 : 푉 푋 × 푉 푋 −→ ℤ → ℤ/2ℤ, where a pair of vertices (푥, 푦) will be declared the same type if (푥, 푦) ∈ ker(휏).

Fix a vertex 푃 in a tree 푋. An 푋-ray from 푃 is an inﬁnite reduced path 훾 = (푒1, 푒2,...) in 푋 with 훼(푒1) = 푃 . Such a path is determined by the corresponding sequence of vertices

푃 = 푃0, 푃1,..., where 푃푛 = 휔(푒푛) = 훼(푒푛+1) for 푛 ≥ 1. We deﬁne an equivalence relation ∼ on 푋-rays by declaring 훾 ∼ 훾′ ⇐⇒ 훾 ∩ 훾′ is an 푋-ray. The equivalence class of an 푋-ray is called an end of the tree 푋. The set of equivalence classes forms a topological space which we denote 퐸푛푑푠(푋) [BL01, 9.1]. For 푋 = 푇푚, a 푚-regular tree with 푚 ≥ 3, it is a fact ([Ser80, I.2.2], [BH10, I.8]) that (the geometric realization of) 퐸푛푑푠(푇푚) is homeomorphic to the Cantor set. Lastly, for a vertex 푃 and an end 휀 = [훾], there exists a unique 푋-ray [푃, 휀), that is, a ray starting at 푃 [BL01, 9.1].

1.2 Quotient Graphs

Recall the deﬁnition of a group action: 8 Deﬁnition 1.2.1. A group 퐺 is said to act on a set 푆 if there is a map

· : 퐺 × 푆 → 푆

(푔, 푎) ↦→ 푔 · 푎

satisfying the following:

1. 1퐺 · 푎 = 푎 for all 푎 ∈ 푆.

2. 푔 · (푕 · 푎) = (푔푕) · 푎 for all 푔, 푕 ∈ 퐺 and 푎 ∈ 푆.

We call a set 푆 a 퐺-set if the group 퐺 acts on 푆. For 푠 ∈ 푆 we set 퐺푠 = {푔 ∈ 퐺 | 푔푠 = 푠}, the

stabilizer of 푠 in 퐺. For 퐴 ⊂ 푆, the stabilizer of 퐴 is 퐺퐴 = {푔 ∈ 퐺 | 푔퐴 = 퐴}. The point-wise

stabilizer or ﬁxator of 퐴 in 퐺 is Fix퐺(퐴) = {푔 ∈ 퐺 | 푔푎 = 푎, ∀푎 ∈ 퐴}.

For a group acting on a graph, we should require that the group action sends vertices to vertices and edges to edges, but moreover, should behave like a graph morphism; that is, there should be no destruction of an edge. Thus, we are led to the following deﬁnition:

Deﬁnition 1.2.2. We say a group 퐺 acts on a graph 푋 if there are group actions of 퐺 on the sets 푉 푋 and 퐸푋 such that 푔 · 훼(푒) = 훼(푔 · 푒) and 푔 · 푒 = 푔 · 푒, for all 푔 ∈ 퐺 and 푒 ∈ 퐸푋. A group 퐺 acts without inversion of edges on a graph 푋 if 푔 · 푒 ̸= 푒 for all 푔 ∈ 퐺 and 푒 ∈ 퐸푋. The action is

called free if 푔 · 푣 ̸= 푣 for all 푣 ∈ 푉 푋 and 푔 ̸= 1퐺 ∈ 퐺. We say 퐺 acts transitively on the vertices (resp., edges) of 푋 if, for all 푣, 푤 ∈ 푉 푋 (푒, 푓 ∈ 퐸푋), there exists a 푔 ∈ 퐺 such that 푔 · 푣 = 푤 (푔 · 푒 = 푓).

Remark 1.2.3. We use the notation 푍 and 푍푛, 푛 ∈ ℤ>1 to denote the inﬁnite cyclic group and cyclic groups of of order 푛, respectively. These groups are of course isomorphic to ℤ and ℤ/푛ℤ.

We use the 푍푛 notation to avoid confusion with the ring of 푝-adic integers, which is generally

written as ℤ푝.

Example 1.2.4. Consider the simple tree 푋 that is just a single segment: 9 {푒, 푒}

푃 푄

The group ℤ/2ℤ = {±1} acts on this tree in an obvious way, by swapping the two vertices and “ﬂipping” the edges:

{푒, 푒}

푃 = (−1) · 푄 푄 = (−1) · 푃

Hence, this action is one with inversion, since (−1) · 푒 = 푒.

Example 1.2.5. Recall that Aut(풞∞) = 퐷∞ (Example 1.1.9). The subgroup generated by the

∼ 푛 automorphism 푏 is ⟨푏⟩ = ℤ. Recall that 푏(푒0) = 푒1 and thus 푏 (푒0) = 푒푛 for any integer 푛. Hence,

⟨푏⟩ acts by translations on 풞∞. This action is transitive on vertices, is free, and is without inversion of edges.

The full automorphism group 퐷∞ acts transitively, but is not free and does act with inversion.

For instance, the automorphism 푎 deﬁned by 푎(푒0) = 푒−1 ﬁxes the vertex 훼(푒0). The automor-

phism 푐 = 푏푎 inverts an edge: 푏푎(푒0) = 푒0.

풞∞ 푒−1 푒0 푒1 -1 0 1 2

In Chapter 2 we will discuss the theory of Bass and Serre regarding group actions on trees. However, this requires groups that act without inversion and therefore is a very necessary assumption in the theory. However, one has no loss in generality by assuming a group acts without inversion of edges on a given graph. Given a group 퐺 acting on a graph 푋, one can always consider the graph 퐵(푋), the barycentric subdivision of 푋. One creates 퐵(푋) from 푋 by subdividing each edge 푒 of 푋 into two edges 푒1, 푒2, and adding a new vertex 푣푒 where 푒 was split. 10 푒 푒1 푒2

푣푒

In this subdivision process, we assume that (푒)2 = 푒1 and (푒)1 = 푒2 and 푣푒 = 푣푒. That is, if we draw both 푒 and 푒 in the above diagram, and then subdivide, one should have the following:

(푒)2 (푒)1 푒 푣푒

푣푒 푒 푒1 푒2

One can now extend an action to 퐵(푋) from an action of 퐺 on 푋 by deﬁning

푔 · 푒1 = (푔 · 푒)1, 푔 · 푒2 = (푔 · 푒)2, 푔 · 푣푒 = 푣푔·푒

One can easily check that this action will be without inversion of edges on 퐵(푋).

Definition 1.2.6. Let 퐺 be a group acting (without inversion) on a graph 푋. Then define the quotient graph 퐺∖푋 by ⋃︁ V(퐺∖푋) = {풪(푣)} 푣∈V 푋 ⋃︁ E(퐺∖푋) = {풪(푒)} 푒∈E 푋 where for 푥 ∈ 푉 푋 ∪ 퐸푋, 풪(푥) = {푔 · 푥| 푔 ∈ 퐺}, the orbit under the action of 퐺. We define 훼(풪(푒)) = 풪(푣) for 푣 ∈ 푉 푋, 푒 ∈ 퐸푋, if there exists a 푔 ∈ 퐺 such that 푔푣 = 훼(푒); and similarly for 휔(풪(푒)). We also require that 풪(푒) = 풪(푒).

Remark 1.2.7. By our deﬁnition of a quotient graph 퐺∖푋, 풪(푒) = 풪(푒). If 퐺 did not act without inversion on 푋, then we could not guarantee that 풪(푒) ̸= 풪(푒), and hence not properly deﬁne a graph. Hence the necessity of having a group act without inversion of edges.

Also note that, by requiring 퐺 to act without inversions, we have that 퐺푒 = 퐺푒,푒 ≤ 퐺훼(푒)

(퐺휔(푒)). 11 Example 1.2.8. In Example 1.2.4 above we had ℤ/2ℤ act with inversion on a segment 푋. So we subdivide and have ℤ/2ℤ act on 퐵(푋):

푒 푒1 푒2 푋 퐵(푋) 푃 푄 푃 푅 푄

Then under this action we have the orbits 풪(푃 ) = {푃, 푄} = 풪(푄), 풪(푒1) = {푒1, 푒2},

풪(푒2) = {푒2, 푒1}, and 풪(푅) = {푅}. Hence, the quotient graph ℤ/2ℤ∖퐵(푋) is a segment,

with edges 풪(푒1) and 풪(푒2) opposites:

풪(푒1) 푍2∖퐵(푋) 풪(푃 ) 풪(푅)

Example 1.2.9. In Example 1.2.5 we saw that the integers act on 풞∞ by translation. Since this

action is transitive on vertices, the quotient graph ℤ∖풞∞ contains a single vertex. We have 풪(푒0) =

{푒푛 | 푛 ∈ ℤ} and 풪(푒0) = 풪(푒0) = {푒푛 | 푛 ∈ ℤ}. Hence the quotient graph is a simple loop:

풪(푒0)

ℤ∖풞∞

풪(0)

Example 1.2.10. Let 퐺 = ℤ/3ℤ and 푋 the graph below.

푋

퐺 acts on 푋 by 120∘ rotations, yielding the quotient graph 12

퐺∖푋

Now suppose 푋 is a graph, 퐺 a group acting without inversion of edges and let 퐺∖푋 be the quotient graph. There is a canonical graph morphism

푝 : 푋 → 퐺∖푋

푥 ↦→ 풪(푥)

called the projection map. We can see in Example 1.2.10 the projection map 푝 : 푋 → 퐺∖푋 for 퐺 = ℤ/3ℤ: 푝 푋 퐺∖푋

The green, blue, and black vertices/edges in 푋 are sent to the corresponding colored vertex or edge in 퐺∖푋. Clearly, 푝 is locally surjective. In fact, if 푥 ∈ 푉 푋 and 푝(푥) = 푦, then

푝|St(푥) : St(푥) → St(푦)

is the quotient map modulo the action of the vertex stabilizer 퐺푥 = {푔 ∈ 퐺| 푔푥 = 푥}. This means

that St(푦) = 퐺푥∖St(푥). To see this, let 푒 ∈ St(푥), so 훼(푒) = 푥. We have

훼(푔푒) = 푔(훼(푒)) = 푔푥. 13

Hence, St(푝(푥)) = {푝(푒) | 푒 ∈ St(푥)}. Moreover, if 푔 ∈ 퐺푥 then 훼(푔푒) = 푥. Hence the orbit of

any 푒 ∈ St(푥) under the action of 퐺푥 must lie in St(푝(푥)). That is, 퐺푥 · 푒 = 풪퐺푥 (푒) = 푝|St(푥)(푒).

Clearly 퐺푒 ≤ 퐺푥. Then we have that for 푒 ∈ St(푥), with 푝(푒) = 푓,

⃒ ⃒ ⃒(︀ )︀−1 ⃒ [퐺푥 : 퐺푒] = ⃒ 푝|St(푥) (푓)⃒ = |퐺푥 · 푒|

and so the index [퐺푥 : 퐺푒] only depends on the projection 푝, and 푓 [BL01, 2.2].

Deﬁnition 1.2.11. Let 푦 be a vertex or edge in a quotient graph 퐺∖푋. Any preimage of 푦 with respect to the projection map 푝 is called a lift of 푦 in 푋. We will denote a lift of 푦 by 푦.˜

Proposition 1.2.12 ([Bog08, Ch2, 1.11]). Let 푒 be an edge in a quotient graph 퐺∖푋. Let 푣 be a lift of 훼(푒). Then there exists a lift 푒˜ of 푒, such that 훼(˜푒) = v.

Proof. This is very easy to see. We simply consider St푋 (푣). The natural projection map 푝 is

locally surjective. That is, 푝 : St푋 (푣) → St퐺∖푋 (훼(푒)) is surjective. Hence, there must exist an

푒˜ ∈ St푋 (푣) that is a lift of 푒.

Using Proposition 1.2.12, one can show the following:

Proposition 1.2.13 ([Bog08, Ch2, 1.12], [Ser80, I.3.1]). Let 퐺 be a group acting without inversion on a connected graph 푋. For any subtree 푇 ′ in 퐺∖푋, there exists a subtree 푇 in 푋 such that

′ 푝|푇 : 푇 → 푇 is an isomorphism.

A subtree 푇 as in the above proposition is also referred to as a lift in 푋 of the subtree 푇 ′.

Hence, we will denote this by 푇 = 푇̃︀′. Lastly, let us deﬁne what is a lift pair as this will be used in Section 2.4.

Deﬁnition 1.2.14 ([Bog08, Ch2, 18.1]). Let 푝 : 푋 → 푌 be a morphism of graphs, where 푋 is a tree and 푌 is connected. Let 푇 be a maximal subtree in 푌 .A lift of the pair of graphs (푇, 푌 ) is a pair of subtrees (푇̃︀ , 푌̃︀) in 푋 that satisfy the following conditions:

1. 푇̃︀ ⊆ 푌̃︀; 14 2. if 푒 ∈ 퐸푌̃︀ ∖ 퐸푇̃︀, then either 훼(푒) or 휔(푒) lies in 푉 푇̃︀;

푝| : 푇 → 푇 푝 퐸푌 퐸푇 퐸푌 퐸푇 3. 푇̃︀ ̃︀ is an isomorphism and maps ̃︀ ∖ ̃︀ bijectively onto ∖ .

Example 1.2.15. Here we give a simple example to illustrate Deﬁnition 1.2.14. The morphism 푝 can be thought of as a reﬂection through the vertex 푥 parallel to the edges 푒 and 푓. The morphism maps those edges as: 푝(푒) = 푝(푓) = 푡.

Let 푇 be a the maximal subtree of 푌 consisting of all edges by 푡. Then 푇̃︀ is the subtree in 푋

indicated in the ﬁgure by the shaded vertices. We deﬁne 푌̃︀ by setting 퐸푌̃︀ = 퐸푋 ∖ {푓}. That is, take all the black vertices of 푋 plus the red vertices to get 푌̃︀. Then one can easily see that (푇̃︀ , 푌̃︀) is a lift of the pair (푇, 푌 ).

푔 푡 푝 푒 푓 푋 푌 푥 푦

1.3 Bruhat-Tits Tree

Here we detail a tree construction that will be very important in the sequel. For any ﬁeld equipped with a discrete valuation, there is a tree structure that can be formed by the set of homotethy equivalence classes of rank two lattices. More generally, one can construct higher dimensional objects called buildings. The trees (or buildings) formed in this way are called Bruhat-Tits trees (buildings) [BT72], [BT84], [AB08], [Wei09]. To get started, we need the following deﬁnitions. Our main reference is Serre [Ser80].

Deﬁnition 1.3.1. Let 퐾 be a ﬁeld A discrete valuation on 퐾 is a surjective map

휈 : 퐾 → ℤ ∪ {∞}

that satisﬁes the following conditions:

1. 휈(푥) = ∞ ⇐⇒ 푥 = 0 15 2. 휈(푥푦) = 휈(푥) + 휈(푦) for all 푥, 푦 ∈ 퐾.

3. 휈(푥 + 푦) ≥ inf{휈(푥), 휈(푦)}

A discrete valuation gives rise to a discrete valuation ring (DVR), the set of elements in 퐾 with non-negative valuation: 풪 = {푥 ∈ 퐾| 휈(푥) ≥ 0}

A uniformizer 휋 ∈ 퐾 is a generator of the unique maximal ideal (휋) = 풪휋. We have 휈(휋) = 1. The ideal (휋) is maximal, hence the quotient 풪/휋풪 is a ﬁeld, called the residue ﬁeld, and will be denoted by 푘.

Remark 1.3.2. Let us collect some facts concerning a DVR. Note that the set of units in 풪, denoted 풪*, consists of all 푥 ∈ 풪 such that 휈(푥) = 0. This follows from the fact that 휈(푥−1) = −휈(푥), and hence if 푥 ∈ 풪, then so is 푥−1. Furthermore, we have that 풪 is a principal ideal domain: every non-zero ideal is of the form (휋푛) = 휋푛풪. So in particular, 풪 is Noetherian, and all ideals of 풪 are of the form (휋푛). We also have that 풪 is a local ring since the ideal generated by 휋 is maximal and unique; it is the ideal containing all the elements with strictly positive valuation:

(휋) = 휋풪 = {푥 ∈ 퐾 | 휈(푥) > 0}.

If 푥 ∈ 풪 and 푥 ̸= 0, then 푥 can be written in the form 푢휋푛 for some 푛 ≥ 0 and 푢 ∈ 풪*. Consequently, 퐾 is the ﬁeld of fractions of 풪 and we can also write, for any 푥 ∈ 퐾*, 푥 = 푢휋푛. Proof of all of the above remarks can be found in [DF04, Section 16.2]. Lastly, notice that for 푥 ∈ 퐾*, since 푥 = 푢휋푛 for some 푛 and 푢 ∈ 풪*, we have:

휈(푥) = 휈(푢휋푛) = 휈(푢) + 푛 · 휈(휋) = 푛. 16 Hence 푥 = 푢휋휈(푥) and we have

풪푥 = 푥풪 = 휋휈(푥)풪 = {푦 ∈ 퐾 | 휈(푦) ≥ 휈(푥)}.

To see this last part, let 푦 ∈ 퐾 with 휈(푦) ≥ 휈(푥). We can write 푦 = 푣휋휈(푦) where 푣 ∈ 풪*. Then since 휈(푦) ≥ 휈(푥), we have 휈(푦) − 휈(푥) ≥ 0 and hence

휈 (︀휋휈(푦)−휈(푥))︀ ≥ 0.

So we have 푦 = (푣휋휈(푦)−휈(푥))휋휈(푥) ∈ 풪휋휈(푥). And clearly, if 푟휋휋(푥) ∈ 풪휋휈(푥), then

휈(푟휋휈(푥)) = 휈(푟) + 휈(푥) ≥ 휈(푥).

Thus 풪휋휈(푥) = {푦 ∈ 퐾 | 휈(푦) ≥ 휈(푥)} and likewise one can show that 휋휈(푥)풪 = {푦 ∈ 퐾 | 휈(푦) ≥ 휈(푥)}.

References for the following examples can be found in [AB08, 6.9] and [DF04, Ch. 16].

Example 1.3.3. Let 퐾 = ℚ, the ﬁeld of rational numbers. Let 푝 be any prime number. Given any 푥 ∈ ℚ, we can always factor 푥 with respect to 푝:

푎 푥 = 푝푛 푏

where 푛, 푎, 푏 ∈ ℤ and 푎 and 푏 are not divisible by 푝. Then we can deﬁne a valuation on ℚ, called the 푝-adic valuation on ℚ, by (︁ 푎)︁ 휈(푥) = 휈 푝푛 := 푛 푏

(Note: This defines a surjective morphism 휈 : ℚ* → ℤ that satisfies (1) and (2) in Defini- tion 1.3.1; we obtain a discrete valuation by setting 휈(0) = ∞.) The valuation ring is 풪(휈) =

{푎/푏 ∈ ℚ | 푝 ̸ | 푏} and the residue ﬁeld 푘 is ℤ/푝ℤ, the ﬁeld of integers modulo 푝. The isomor-

−1 phism 풪(휈) → ℤ푝 is given by 푎/푏 ↦→ (푎 mod 푝)(푏 mod 푝) . 17 Example 1.3.4. Consider 퐾 = 푘(푡), the rational function ﬁeld over a ﬁeld 푘. Fix a point 푎 ∈ 푘. By unique factorization, for any non-zero polynomial 푓(푡) we can write:

푛 푓(푡) = (푡 − 푎) 푓0(푡)

where 푓0 has constant term ̸= 0 (i.e., (푡 − 푎) does not divide 푓0(푡)). We deﬁne a valuation with respect to 푎 on 퐾 by: (︂ )︂ (︂ )︂ 푓(푡) 푛 푓0(푡) 휈푎 = (푡 − 푎) := 푛 푔(푡) 푔0(푡)

where 푓(푡), 푔(푡) ∈ 푘[푡] and 푔(푡) ̸= 0. This is called the “valuation at 푎.” The corresponding DVR is

풪푎 = {퐹 ∈ 푘(푡) | 퐹 (푎) is defined}, with uniformizer element 휋 := 푡 − 푎. The residue field is 푘[푡]/(푡 − 푎), as can be seen by an isomorphism 풪푎/(푡 − 푎)풪푎 → 푘[푡]/(푡 − 푎) defined by 퐹 ↦→ 퐹 (푎). Note that for 퐹 ∈ 퐾, if

휈푎(퐹 ) ≥ 0 then 휈푎(퐹 ) is the order of the zero of 퐹 at 푎; if 휈푎(퐹 ) < 0 then 휈푎(퐹 ) is the order of the pole of 퐹 at 푎. As a special case, we emphasize when 푎 = 0, the “valuation at 0,” which in the sequel we will ∼ denote 휈0 = 휈+. Here the uniformizer element is 휋 = 푡 and the residue ﬁeld is 푘[푡]/(푡) = 푘.

Example 1.3.5. Consider again 퐾 = 푘(푡). Above we saw that one can deﬁne many discrete valuations on 퐾, with each being a “valuation at 푎 ∈ 푘.” We have a different valuation that can be formed on 퐾 which can be thought of as the “valuation at inﬁnity,” if one considers 퐾 as the

function ﬁeld of the projective line ℙ1(푘). It is deﬁned as follows:

(︂푓 )︂ 휈 = deg(푔) − deg(푓) ∞ 푔

where 푓, 푔 ∈ 푘[푡], 푔 ̸= 0, and deg is the degree of the polynomial. 푓 Let 풪 be the corresponding valuation ring. Then we see that, if 퐹 = for 푓, 푔 ∈ 푘[푡], 푔 ̸= 0, ∞ 푔 18 and 퐹 ∈ 풪∞,

휈∞(퐹 ) = deg(푔) − deg(푓) ≥ 0 and thus deg(푔) ≥ deg(푓). So

{︂ ⃒ }︂ 푓 ⃒ 풪∞ = ∈ 퐾⃒ deg(푔) ≥ deg(푓) . 푔 ⃒

1 We choose uniformizer element = 푡−1 and can, similar to the previous example, identify the 푡

residue ﬁeld with 푘 via 퐹 ↦→ 퐹 (∞). In the sequel, we will denote 휈∞ = 휈−.

Example 1.3.6. For one last example, we let 퐾 = 푘((푡)), the field of formal Laurent series with coefficients in the field 푘. One can define a discrete valuation on 퐾 by

(︃ ∞ )︃ ∑︁ 푖 휈 푎푖푡 = 푛. 푖≥푛

The corresponding DVR is 풪휈 = 푘[[푡]], the ring of formal power series in 푡 over 푘, with uniformizer element 휋 = 푡. The residue ﬁeld is 푘.

Lattices

For the remainder of this section, we ﬁx 퐾 a ﬁeld with discrete valuation 휈, discrete valuation ring 풪, and uniformizer 휋.

Deﬁnition 1.3.7. Call 퐿 a lattice in a 퐾-vector space 푉 if 퐿 is an 풪-submodule in 푉 that also generates the 퐾-vector space 푉 . Note that a lattice 퐿 in 푉 is, in particular, a free 풪-module of rank 2.

From now on, 푉 will denote a vector space of dimension 2 over 퐾. Given a basis {푒1, 푒2} for 푉 , we have

퐿 = {푎푒1 + 푏푒2|푎, 푏 ∈ 풪} = 풪푒1 ⊕ 풪푒2 19 a lattice in 퐿. In fact, it can be shown that all lattices are of this form ([Gar97, Sec. 18.4]). We will

2 call the lattice 퐿 = 풪푒1 ⊕ 풪푒2 = 풪 the standard lattice.

Equivalence Classes

There is an action of 퐾* on the set of lattices in 푉 . If 푥 ∈ 퐾* and 퐿 is a lattice, then 푥퐿 is also a lattice. For instance, suppose we have 퐿 = 풪푒1 ⊕ 풪푒2 for {푒1, 푒2} a basis for 푉 . Then since 푥풪 = 풪푥 we have:

푥퐿 = 풪(푥푒1) ⊕ 풪(푥푒2).

Since {푒1, 푒2} is a basis for 푉 , so is {푥푒1, 푥푒2} and hence 푥퐿 is a lattice in 푉 . Thus we can deﬁne an equivalence relation on the set of lattices:

Deﬁnition 1.3.8. We say two lattices 퐿 and 퐿′ are (homotethy) equivalent if there exists 푥 ∈ 퐾* such that 퐿푥 = 퐿′. We will write 퐿 ∼ 퐿′ to signify equivalence. Denote the equivalence class of

퐿 by Λ or [퐿]풪 = [퐿]. Set 푋 = {[퐿] | 퐿 is a lattice in 푉 }

the set of all lattice classes.

We will also use the notation [퐿] = [[푓1, 푓2]] or [퐿] = [⟨푓1, 푓2⟩] for 퐿 = 풪푓1 ⊕ 풪푓2.

The Tree of Lattice Classes

We now seek to define a combinatorial tree structure on 푋. The set of vertices will be equal to 푋, and edges will be defined through a distance function defined on lattice classes. To define a distance, we need the following lemma.

′ ′ Lemma 1.3.9. Let 퐿, 퐿 be lattices in 푉 . Then there exists an 풪-basis {푓1, 푓2} for 퐿 such that 퐿

* has 풪-basis {푥1푓1, 푥2푓2} for scalars 푥1, 푥2 ∈ 퐾 . More precisely, there exists integers 푎, 푏 ∈ ℤ

푎 푏 ′ such that {휋 푓1, 휋 푓2} is an 풪-basis for 퐿 .

Proof. Firstly, note that we can choose an integer 푀 large enough so that 휋푀 퐿′ ⊂ 퐿. To see this, 20 ′ suppose {푒1, 푒2} is an 풪-basis for 퐿 and {푔1, 푔2} is an 풪-basis for 퐿 . Then we can write

푔1 = 푥푒1 + 푦푒2, 푔2 = 푧푒1 + 푤푒2,

where 푥, 푦, 푧, 푤 ∈ 퐾, since both 풪-bases generate 푉 as a 퐾 vector space. Then we set

푚1 = − min(휈(푥), 휈(푦))

푚2 = − min(휈(푧), 휈(푤))

푀 = max(푚1, 푚2).

Then 휈(푠휋푀 ) ≥ 0 for 푠 = 푥, 푦, 푧, 푤. Hence,

푀 푀 푀 휋 푔1 = (휋 푥)푒1 + (휋 푦)푒2 ∈ 퐿

푀 푀 푀 휋 푔2 = (휋 푧)푒1 + (휋 푤)푒2 ∈ 퐿.

Therefore, we have 휋푀 퐿′ ⊂ 퐿. Recall that 풪 is a principal ideal domain. Hence, by the theory of ﬁnitely generated modules

over PID’s (cf. [DF04, 12.1, Thm 4]), we have that there exists an 풪-basis {푓1, 푓2} for 퐿 and

* 푀 ′ scalars 푥1, 푥2 ∈ 퐾 such that {푥1푓1, 푥2푓2} is a basis for 휋 퐿 , where 푥1|푥2.

Now recall that every element in 푦 ∈ 퐾* can be written in the form 푦 = 푢휋푡 where 푡 ∈ ℤ

* ′ and 푢 ∈ 풪 . In particular, 푡 = 휈(푦). Writing 푥1 and 푥2 in this way, we see that 퐿 has 풪-basis

푎 푏 {휋 푓1, 휋 푓2} where 푎 = 푀 + 휈(푥1) and 푏 = 푀 + 휈(푥2).

Lastly, since 푥1|푥2, we have that 푎 ≤ 푏.

Before the next lemma, let us note that the lattices 퐿 and 퐿′ satisfy 퐿′ ⊂ 퐿 if and only if the integers 푎 and 푏 from Lemma 1.3.9 are both non-negative. Indeed, if 퐿′ ⊂ 퐿, then using the bases above we have

′ 푎 푏 퐿 = 풪(휋 푓1) ⊕ 풪(휋 푓2) ⊂ 풪푓1 ⊕ 풪푓2 = 퐿. 21 Clearly this is if and only if 휋푎, 휋푏 ∈ 풪 ⇐⇒ 푎 and 푏 ≥ 0. Secondly, let us note that, if 퐿′ ⊂ 퐿, then the quotient 풪-module 퐿/퐿′ is isomorphic to:

퐿/퐿′ ∼= (풪/(휋푎)) ⊕ (︀풪/(휋푏))︀ .

′ 푎 푏 Lemma 1.3.10. Let 퐿 and 퐿 be lattices with 풪-basis {푓1, 푓2} and {휋 푓1, 휋 푓2}, respectively from Lemma 1.3.9. The number |푎 − 푏| only depends on the lattice classes of 퐿 and 퐿′.

* ′ ′ Proof. For scalars 푥, 푦 ∈ 퐾 we have 퐿 ∼ 푥퐿 and 퐿 ∼ 푦퐿 . Then 푥퐿 has basis {푥푓1, 푥푓2} and

′ 푎+휈(푦) 푏+휈(푦) 푦퐿 has basis {휋 푓1, 휋 푓2}. Note that

휋푎+휈(푦) = 휋푎+휈(푦)−휈(푥)휋휈(푥),

푏+휈(푦) −1 ′ 푎+푐 푏+푐 and similarly for 휋 . Hence, setting 푐 = 휈(푦푥 ), we have that 푦퐿 has 풪-basis {휋 푥푓1, 휋 푥푓2}. Subtracting the powers of 휋 in absolute value, we have |푎 + 푐 − (푏 + 푐)| = |푎 − 푏|.

We can now deﬁne a distance between lattice classes:

Deﬁnition 1.3.11. Let Λ and Λ′ be two lattice classes in 푋. Then the distance between Λ and Λ′ is given by 푑(Λ, Λ′) = |푎 − 푏|

where 푎 and 푏 are the integers deﬁned in Lemma 1.3.9. We will say that two lattice classes Λ, Λ′ are adjacent if 푑(Λ, Λ′) = 1.

It is straightforward that 푑(Λ, Λ′) = 0 ⇐⇒ Λ = Λ′. Suppose we have 푑(Λ, Λ′) = 1. Then we can choose lattice representatives 퐿 and 퐿′ such that

′ 퐿 has basis {푓1, 푓2} and 퐿 has basis {휋푓1, 푓2}. In particular we have

휋퐿 ⊂ 퐿′ ⊂ 퐿.

In fact, adjacency is symmetric since if we have 휋퐿 ⊂ 퐿′ ⊂ 퐿, then we also have 휋퐿′ ⊂ 휋퐿 ⊂ 퐿′ 22 and 휋퐿 ∼ 퐿. Hence if 푑(Λ, Λ′) = 1 then 푑(Λ′, Λ) = 1. Also, if we have 휋퐿 ⊂ 퐿′ ⊂ 퐿 then notice that

퐿/퐿′ ∼= 풪/풪휋 ∼= 푘,

where recall 푘 was called the residue ﬁeld associated to the DVR 풪. Expanding upon the above, if Λ and Λ′ are lattice classes and 푑(Λ, Λ′) = 푛, then we can ﬁnd

′ ′ 푛 ′ lattices 퐿 ∈ Λ, 퐿 ∈ Λ such that {푓1, 푓2} is a basis for 퐿 and {휋 푓1, 푓2} is a basis for 퐿 . So in particular, we have 퐿′ ⊂ 퐿 and

퐿/퐿′ ∼= 풪/풪휋푛.

In fact, if we are given 퐿 ∈ Λ, then in every other lattice class there exists a unique lattice 퐿′ satisfying the above (but with varying 푛). We prove this below:

Proposition 1.3.12. Let 퐿 ∈ Λ be given. Then in each class Λ′ ∈ 푋, there exists a unique lattice 퐿′ such that 퐿′ ⊂ 퐿 and

퐿/퐿′ ∼= 풪/풪휋푛,

where 푛 = 푑(Λ, Λ′).

Proof. Let Λ′ ∈ 푋 and set 푛 = 푑(Λ, Λ′). Suppose 퐿′ and 퐿′′ are two lattices in Λ′ such that 퐿′ ⊂ 퐿

with 퐿/퐿′ ∼= 풪/풪휋푛 and 퐿′′ ⊂ 퐿 with 퐿/퐿′′ ∼= 풪/풪휋푛. Then by Lemma 1.3.9 we know there

′ 푛 exists an 풪-basis {푓1, 푓2} for 퐿 such that 퐿 has basis {휋 푓1, 푓2}; and there exists a basis {푒1, 푒2}

′′ 푛 for 퐿 such that 퐿 has basis {휋 푒1, 푒2}. Since both lattices 퐿′ and 퐿′′ lie in Λ′, they are equivalent; hence there exists 휆 ∈ 퐾* such that 휆퐿′ = 퐿′′. That is,

푛 푛 풪(휆휋 푓1) ⊕ 풪(휆푓2) = 풪(휋 푒1) ⊕ 풪(푒2).

Writing 휆 = 푢휋푘 for some 푢 ∈ 풪* and 푘 ∈ ℤ we have

′ 푘+푛 푘 푛 휆퐿 = 풪(휋 푓1) ⊕ 풪(휋 푓2) = 풪(휋 푒1) ⊕ 풪(푒2). 23 ′ ′′ But this holds if and only if 푘 = 0 and 푓1 = 푒1, 푓2 = 푒2. Hence 퐿 = 퐿 .

Now deﬁne a tree structure on 푋. We deﬁne vertices by 푉 푋 = 푋. For each Λ, Λ′ ∈ 푋, if

′ ′ 푑(Λ, Λ ) = 1, we deﬁne an edge 푒ΛΛ′ , with 훼(푒ΛΛ′ ) = Λ and 휔(푒ΛΛ′ ) = Λ . We set 푒ΛΛ′ = 푒Λ′Λ.

Then 퐸푋 = {푒ΛΛ′ | Λ, Λ ∈ 푋}.

Theorem 1.3.13. X is a tree.

Before we prove this, let us recall a few deﬁnitions from module theory (cf. [DF04] and [Hun80]).

Deﬁnition 1.3.14. Let 푅 be a ring, 푀 an 푅-module. 푀 is called simple or irreducible if there are no proper, nonzero submodules of 푀. Thus if 푀 is simple and 푁 ⊆ 푀, then either 푁 = 푀 or 푁 = 0.

A normal series for an 푅-module 푀 is a ﬁnite chain of submodules: 0 = 푀0 ⊂ 푀1 ⊂ · · · ⊂

푀푛 = 푀. The quotients 푀푖+1/푀푖 are called factors. The length of the series is 푛, the number of non-trivial factors, and is an invariant of 푀. A normal series is called a composition series if each factor is a simple module.

Proof of Theorem 1.3.13. First we must show that 푋 is connected. Let Λ, Λ′ ∈ 푋 with 푑(Λ, Λ′) =

푛. Let 퐿, 퐿′ be the respective lattice class representatives such that 퐿′ ⊂ 퐿 and 퐿/퐿′ ∼= 풪/풪휋푛. We want to show that there exists a path in 푋 between Λ and Λ′.

Note that the 풪-module 퐿/퐿′ ∼= 풪/풪휋푛 is a cyclic module. By the Fourth Isomorphism Theorem for modules, 풪-submodules of 풪/풪휋푛 correspond to 풪-submodules of 풪휋푛. But there are only ﬁnitely many such submodules. Hence, there exists a composition series for 퐿/퐿′:

′ 0 = 푀0 ⊂ 푀1 ⊂ · · · ⊂ 푀푛 = 퐿/퐿 .

By the Fourth Isomorphism Theorem again, each submodule 푀푖 corresponds to a submodule 퐿푖 in 퐿 that contains 퐿′. Hence, we have a composition series:

′ 퐿 = 퐿푛 ⊂ 퐿푛−1 ⊂ · · · ⊂ 퐿0 = 퐿. 24

Since each factor 푀푖+1/푀푖 is simple, so is each factor 퐿푖/퐿푖+1, 0 ≤ 푖 ≤ 푛 − 1. It follows from the theory on simple modules that the 풪-modules 퐿푖/퐿푖+1 are each isomorphic to 풪/m, where m is a maximal ideal in 풪. But because 풪 is a DVR, there is only one maximal ideal, the ideal generated by the uniformizer element 휋. Hence, we have for each 푖 that

∼ 퐿푖/퐿푖+1 = 풪/풪휋 = 푘,

the residue ﬁeld. Therefore, if we denote lattice classes of the 퐿푖 by [퐿푖] = Λ푖 for 0 ≤ 푖 ≤ 푛, then

′ we see that Λ = Λ0, Λ1,..., Λ푛 = Λ gives a sequence of adjacent vertices in 푋. Thus, by taking the corresponding edges, we have a path from Λ to Λ′ in 푋. So 푋 is connected. To show that 푋 is a tree, we must now show that there are no closed, reduced, non-trivial paths in 푋. Let 푙 = 푒1 . . . 푒푛, 푛 ≥ 1 be a reduced, non-trivial path in 푋. Denote by 푥0, 푥1, . . . , 푥푛 the vertices from the path 푙, where 푥푖 = 훼(푒푖 + 1), 0 ≤ 푖 ≤ 푛 − 1 and 푥푛 = 휔(푒푛). We must show that

푥0 ̸= 푥푛, else we will have a loop.

To show this, it sufﬁces to show that 푑(푥0, 푥푛) = 푛 (since lattice classes are equal if and only if their distance is zero). We show this by induction on 푛, the length of the path.

For the case 푛 = 1, we know that 푥0 and 푥푛 = 푥1 are adjacent and so by deﬁnition we have

푑(푥0, 푥1) = 1. Now suppose 푛 ≥ 2. From our proof of connectedness above, we know that we can choose ∼ lattices 퐿푖 ∈ 푥푖, 0 ≤ 푖 ≤ 푛 such that 퐿푖+1 ⊂ 퐿푖 and 퐿푖/퐿푖+1 = 풪/풪휋;

퐿푛 ⊂ 퐿푛−1 ⊂ · · · ⊂ 퐿0

By the inductive hypothesis, we know that 푑(푥0, 푥푛−1) = 푛 − 1. In particular, 퐿푛−1 ⊂ 퐿0

∼ 푛−1 and 퐿0/퐿푛−1 = 풪/풪휋 . The inductive hypothesis also tells us that 푥0 and 푥푛 are not adjacent. Hence, we know

휋퐿0 ⊈ 퐿푛−1 ⊂ 퐿0 25 and

퐿푛−1 ⊈ 휋퐿0.

To show that 푥0 and 푥푛 are not adjacent, we must show that 퐿푛 ⊈ 휋퐿0. One way to show this is to show that 퐿푛 is not trivial modulo 휋퐿0; since if we did have 퐿푛 ⊂ 휋퐿0, then 퐿푛 would be trivial in the quotient 퐿/휋퐿0.

By our choice of 퐿푖’s, we have that 휋퐿푛−1 ⊂ 퐿푛−1 ⊂ 퐿푛−2 and 휋퐿푛−1 ⊂ 퐿푛 ⊂ 퐿푛−1. But we also have

휋퐿푛−1 ⊂ 퐿푛−1 ⊂ 퐿푛−1, by multiplying the ﬁrst relation by 휋. Hence, via the Fourth Isomorphism Theorem, the lattices

∼ 2 휋퐿푛−2 and 퐿푛 correspond to submodules of 퐿푛−1/휋퐿푛−1 = 푘 (i.e. a 푘-plane). These submodules must correspond to lines in 퐿푛−1/휋퐿푛−1 since 휋퐿푛−2 and 퐿푛 are not trivial and neither is equal to 퐿푛−1. Moreover, these lines must be distinct, else we would have 휋퐿푛−2 ∼ 퐿푛, meaning the vertices 푥푛−2, 푥푛−1, 푥푛 = 푥푛−2 in the path 푙 would form a backtracking; but this is impossible because we assumed 푙 was reduced.

Hence, we have that 퐿푛−1 = 퐿푛 + 휋퐿푛−2. If we take this modulo 휋퐿0, we get

퐿푛−1 ≡ 퐿푛 (mod 휋퐿0)

since 퐿푛−2 ⊂ 퐿0 and thus 휋퐿푛−2 ⊂ 휋퐿0. From our inductive hypothesis, 퐿푛−1 ⊈ 휋퐿0; thus

퐿푛 ⊈ 휋퐿0 as well.

Remark 1.3.15. The above proof shows that the distance function 푑(Λ, Λ′) (Deﬁnition 1.3.11)

′ coincides with the combinatorial distance function 푑푋 (Λ, Λ ) (Deﬁnition 1.1.16). 26 Spheres and Projective Lines

Let us examine the local structure of a vertex in 푋. Recall we have the star of a vertex, the set of all edges which lie on said vertex (say 푣 ∈ 푉 푋):

St푋 (푣) = {푒 ∈ 퐸푋 | 푣 = 훼(푒)}.

We deﬁne spheres and balls in 푋 by means of the combinatorial distance function.

Deﬁnition 1.3.16. For any vertex 푣 of 푋 , we shall denote the sphere and ball of radius 푛 ∈ ℤ≥0 around 푣 as

푛 푛 ′ ′ 햲푋 (푣) = 햲 (푣) = {푣 ∈ 푉 푋 | 푑푋 (푣, 푣 ) = 푛},

푛 푛 ′ ′ 햡푋 (푣) = 햡 (푣) = {푣 ∈ 푉 푋 | 푑푋 (푣, 푣 ) ≤ 푛}.

1 ∼ 1 We will show that 햲 (푣) = ℙ (푘), the projective line over 푘; in fact we will show that the vertices at distance 푛 from 푣 are given by

푛 ∼ 1 푛 햲 (푣) = {푦 ∈ 푉 푋 | 푑푋 (푣, 푦) = 푛} = ℙ (풪/풪휋 ), where the last object is the projective line of the ring 풪/풪휋푛. This is deﬁned in the following:

Definition 1.3.17. For a commutative ring 푅 (with 1), let ∼ be an equivalence relation on 푅 × 푅 defined by: (푎, 푏) ∼ (푐, 푑) ⇐⇒ ∃푢 ∈ 푅* such that (푢푎, 푢푏) = (푐, 푑). The projective line over 푅 is defined to be the set of equivalence classes of pairs of relatively prime elements of 푅:

ℙ(푅) :={(푎, 푏) ∈ 푅 × 푅 | (푎) + (푏) = 푅}/ ∼

={[(푎, 푏)] | (푎, 푏) ∈ 푅 × 푅 and 푎 is relatively prime to 푏} where [(푎, 푏)] = {(푐, 푑) ∈ 푅 × 푅 | 푐푅 + 푑푅 = 푅; ∃푢 ∈ 푅* such that(푢푐, 푢푑) = (푎, 푏)}. 27 Remark 1.3.18. Recall that, if 푅 is a PID, then 푎, 푏 ∈ 푅 are said to be relatively prime if the ideal generated by 푎 and 푏 is trivial, i.e. (푎, 푏) = 1. More generally, the notion of being “relatively prime” is replaced by that of ideals being comaximal: (푎) + (푏) = 푅. In a PID, these two notions are equivalent (but not in general). But so for general rings, one can deﬁne two elements 푎 and 푏 to be relatively prime if the ideals (푎) and (푏) are comaximal, which we did for the above deﬁnition.

Lemma 1.3.19. The units in 풪/(휋푛) are exactly the classes 푥 = 푥 + 풪휋푛 where 휈(푥) = 0 (i.e. where 푥 ∈ 풪*).

Proof. Let 푥 ∈ 풪/(휋푛) and suppose 휈(푥) = 0. Since 푥 ∈ 풪 is a unit of 풪, there exists an element 푥−1 ∈ 풪 such that 푥푥−1 = 1. So for any 푓 ∈ (휋푛), we have

(푥 + 푓)푥−1 = 푥푥−1 + 푓푥−1 = 1 + 푓푥−1.

Hence, modulo (휋푛) we have 푥 · 푥−1 = 1 and so 푥 is a unit in 풪/(휋푛). Now conversely, suppose 푥 ∈ 풪/(휋푛) is a unit. Assume 휈(푥) = 푘 > 0. Then 휈(푥 · 휋푛−푘) =

푘 + 푛 − 푘 = 푛, and hence 푥 · 휋푛−푘 ∈ (휋푛). But then recall that every element 푧 of 핂 can be written 푧 = 푢휋휈(푧), where 푢 ∈ 풪*. Thus,

푥휋−푛−푘 = 푢휋휈(푥휋푛−푘) = 푢휋푛,

and hence (푥)(휋푛−푘) = 0, meaning 푥 is a zero-divisor. No unit can be a zero-divisor, so this is a contradiction and therefore we must have 휈(푥) = 0.

By the above lemma, we know what are the units in 풪/(휋푛). By deﬁnition of the projective line:

1 푛 푛 ℙ (풪/(휋 )) = {[푎, 푏] | (푎) + (푏) = 풪/(휋 )}.

It is clear that if either of the elements 푎 or 푏 is a unit, then the ideals 푎풪 and 푏풪 are co-maximal. Suppose instead that neither 푎 nor 푏 is a unit in 풪/(휋푛). By deﬁnition of the quotient ring, we can 28 write 푎 = 훼 + (휋푛) and 푏 = 훽 + (휋푛) for some 훼, 훽 ∈ 풪. Since neither 푎 nor 푏 is a unit, neither is 훼 or 훽 in 풪. Hence 휈(훼) > 0 and 휈(훽) > 0. It follows by deﬁnition of a discrete valuation that:

휈(푎 + 푏) ≥ min(휈(훼), 휈(훽)) > 0.

Thus, (푎) + (푏) ⊆ (푡) ⸀ 풪/(휋푛). So necessarily, we must have for [푎, 푏] ∈ ℙ1(풪/(휋푛)) that at least one of 푎 or 푏 is a unit in 풪/(휋푛).

Proposition 1.3.20. The points of the projective line ℙ1(풪/풪휋) correspond bijectively to the vertices on a sphere 햲1(푣), for any vertex 푣 ∈ 푉 푋.

Proof. Let 퐿0 be a lattice of 푉 in the lattice class 푣 = [퐿0]. If 푥 is a vertex with 푑푋 (푣, 푥) = 푛,

∼ 푛 then there exists a unique lattice 퐿 ∈ [퐿] = 푥 such that 퐿 ⊂ 퐿0 and 퐿0/퐿 = 풪/풪휋 . Moreover,

푛 since 휋 퐿0 ⊂ 퐿 ⊂ 퐿0, we have that

푛 푛 푛 푛 ∼ 푛 푛 휋 퐿0/휋 퐿0 ⊂ 퐿/휋 퐿0 ⊂ 퐿0/휋 퐿0 = 풪/풪휋 × 풪/풪휋 .

푛 푛 푛 ∼ 푛 This implies that 퐿0/휋 퐿0 is a free rank 2 풪/풪휋 -module and, since 퐿/휋 퐿0 = 풪/풪휋 , we have that this is a direct factor of rank 1. These are precisely the points of the projective line

푛 ∼ 1 푛 푛 푛 ℙ(퐿0/휋 퐿0) = ℙ (풪/풪휋 ). Moreover, the 풪/풪휋 -module 퐿/휋 퐿0 is in fact a projective mod-

푛 ule, as it is a direct summand (direct factor) of the free module 퐿0/휋 퐿0.

1 ∼ 1 ∼ 1 Hence, if 푛 = 1, we have that 햲 (푣) = ℙ (풪/풪휋) = ℙ (푘).

From the above proposition we see that, if 푘 is a ﬁnite ﬁeld of order 푞, then |St푋 (푣)| = 푞 + 1. It follows then that the tree Bruhat-Tits 푋 is regular of degree 푞 + 1.

Example 1.3.21. Consider the Bruhat-Tits tree 푋 for the valuation 휈+ on 퐾 = 헄(푡). Then we have 29 the following correspondences in the projective line ℙ1(풪/풪푡푛):

푛 [1: 0] ↔ [⟨푒1, 푡 푒2⟩]

푛 [1: 1] ↔ [⟨푒1 + 푒2, 푡 푒2⟩]

푛 [0: 1] ↔ [⟨푒2, 푡 푒1⟩]

푛−1 푛−1 푛 [1: 푏0 + 푏1푡 + ··· 푏푛−1푡 ] ↔ [⟨푒1 + (푏0 + 푏1푡 + ··· + 푏푛−1푡 )푒2, 푡 푒1⟩]

푛−1 푛−1 푛 [푎1푡 + ··· 푎푛−1푡 : 1] ↔ [⟨(푎1푡 + ··· 푎푛−1푡 )푒1 + 푒2, 푡 푒1⟩],

where 푏0, . . . , 푏푛−1, 푎0, . . . , 푎푛−1 ∈ 헄.

Completions

Let us mention here the completion of a discrete valuation.([AB08, 6.9], [Ser80, II.1.1])

Let 퐾 be a ﬁeld with discrete valuation 휈. Fix a non-zero constant 휆 ∈ ℝ with |휆| > 1. Then we deﬁne a real-valued absolute value function on 퐾 by

−휈(푥) |푥|휆 := 휆 ,

where 푥 ∈ 퐾. This induces a metric: 푑휆(푥, 푦) = |푥 − 푦|휆 since the absolute value satisﬁes the ultra-metric inequality: |푥 + 푦| ≤ max{|푥|, |푦|}.

If the ﬁeld 퐾 is not complete, then we can form the completion 퐾̂︀ via the metric 푑휆. Via continuity,

we can pass the ﬁeld operations of 퐾 to 퐾̂︀. Likewise, we can deﬁne a discrete valuation 휈̂︀ : 퐾̂︀ → ℤ ∪ {∞} by

휈(푥) = lim 휈(푥푛) ̂︀ ̂︀ 푛→∞

where 푥푛 → 푥̂︀ in 퐾,̂︀ (푥푛) a sequence in 퐾. Then the DVR for 휈̂︀ can be defined as the closure 풪̂︀ of 풪 in 퐾̂︀; or, we can define 풪̂︀ via an inverse limit (cf. Definition 3.2.4): 풪̂︀ = lim 풪/풪휋푛. The ←− residue field is again 푘, the same one for 퐾 and 풪. 30 Since 퐾̂︀ is a field with discrete valuation, we can produce a tree 푋̂︀ just like we did for 퐾. Set ∼ 푉̂︀ = 푉 ⊗퐾 퐾̂︀. This is a two-dimensional 퐾̂︀ vector space, since 푉 = 퐾 ⊕ 퐾 and thus:

∼ (︁ )︁ (︁ )︁ 푉̂︀ = 푉 ⊗퐾 퐾̂︀ = (퐾 ⊕ 퐾) ⊗퐾 퐾̂︀ = 퐾 ⊗퐾 퐾̂︀ ⊕ 퐾 ⊗퐾 퐾̂︀ .

∼ ∼ It follows that 푉̂︀ = 퐾̂︀ ⊕ 퐾̂︀ since 퐾 ⊗퐾 퐾̂︀ = 퐾̂︀. Then we can deﬁne lattices, lattice classes, etc. for the 퐾̂︀-vector space 푉̂︀, and hence obtain a tree 푋̂︀. But it turns out that the tree 푋 of the 퐾 vector space 푉 is isomorphic to the tree 푋̂︀. If 퐿 is a lattice in 푉 , we associate to it 퐿̂︀ = 퐿 ⊗풪 풪̂︀, which is a lattice in 푉̂︀ (one can show this is a lattice by the same way we showed 푉̂︀ is a vector space). This yields a bijection of the set of lattices of 푉 with the set of lattices of 푉̂︀, and hence induces an isomorphism between 푋 and 푋̂︀.

Example 1.3.22. Recall Example 1.3.3 of the 푝-adic valuation on ℚ. The residue ﬁeld is ℤ/푝ℤ, a ﬁeld of order 푝. Hence, the Bruhat-Tits tree 푋 constructed by the 푝-adic valuation on ℚ is a regular tree of degree 푝 + 1; that is, every vertex 푥 has deg(푥) = 푝 + 1.

−휈푝(푥) The completion of ℚ induced by the 푝-adic absolute value |푥|푝 = 푝 is the ﬁeld of 푝-adic numbers ℚ푝. The valuation ring 풪̂︀ is ℤ푝, the ring of 푝-adic integers. As mentioned above, the tree

푋 and 푋̂︀ are isomorphic. Hence, the Bruhat-Tits tree for both ℚ and ℚ푝 is a 푝 + 1-regular tree,

푇푝+1.

Example 1.3.23. Let 퐾 = 푘(푡), the field of rational functions over the field 푘 = 픽푞, a finite field of order 푞. The valuations 휈+ defined in Example 1.3.4 and 휈− defined in Example 1.3.5 on 퐾 both give rise to Bruhat-Tits trees 푇 + and 푇 −. These trees are both regular of degree 푞 + 1.

If we complete 퐾 = 픽푞(푡) with respect to the valuation 휈+, then the completed field can be identified with the field of formal Laurent series 픽푞((푡)), and the valuation 휈̂︁+ is equivalent to the valuation in Example 1.3.6, with DVR 픽푞[[푡]].

On the other hand, completing 퐾 with respect to the valuation 휈−, we obtain a completed ﬁeld

−1 −1 that can be identiﬁed with 픽푞((푡 )), the ﬁeld of Laurent polynomials in indeterminate 푡 , and

−1 the DVR is 픽푞[[푡 ]]. 31 1.4 Twin Trees

In this section we deﬁne twin trees and show how two Bruhat-Tits trees can be “twinned.” Our main reference is the paper [RT94], which is the genesis on twin trees. If 푇 is a tree, recall from

Deﬁnition 1.1.16 that we can deﬁne a function 훿 : 푉 푇 × 푉 푇 → ℤ>0 by

훿(푥, 푦) = 푛

where 푛 is the length of the unique geodesic from 푥 to 푦. One can check that

1. 훿(푥, 푦) = 0 ⇐⇒ 푥 = 푦;

2. 훿(푥, 푦) = 훿(푦, 푥);

3. 훿(푥, 푦) = 1 ⇐⇒ 푥 and 푦 are adjacent.

4. If 푚 = 훿(푥, 푦), then for any 푦′ adjacent to 푦, 훿(푥, 푦′) = 푚±1; and if 푚 > 0, then ∃ a unique 푦′ such that 훿(푥, 푦′) = 푚 − 1.

Conversely, suppose 푋 is a set endowed with a function 훿 : 푋 ×푋 → ℤ≥0 satisfying the above properties (1), (2), and (4). Then 푋 is a tree in the sense of Definition 1.1.13; one sets 푉 푋 = 푋 and define edges for all pairs of vertices satisfying (3). Now we consider two trees 푇 + and 푇 −. We can define a function from 푉 푇 + to 푉 푇 − that is “dual” to property (4) above. This allows us to define a “twin tree.”

Deﬁnition 1.4.1. A twin tree is a tuple 푇 = (푇 +, 푇 −, 훿*) where (푇 +, 훿+) and (푇 −, 훿−) are trees with 훿+, 훿− combinatorial distance functions; and 훿* is a function on pairs of vertices of opposite

+ − signs in 푉 푇 ⊔ 푉 푇 with values in ℤ≥0, which satisﬁes the following condition:

(codist)

1. For any 푥+ ∈ 푇 +, 푦− ∈ 푇 −, we have 훿*(푥+, 푦−) = 훿*(푦−, 푥+) 32 2. Set 푚 = 훿*(푥+, 푦−). For each 푦′− ∈ 푉 푇 − adjacent to 푦−, we have 훿*(푥+, 푦−) = 푚±1. If 푚 > 0, then there exists a unique 푦′− such that 훿*(푥+, 푦−) = 푚 + 1. Similarly, this is true if we interchange 푇 + and 푇 −.

If (codist) is satisﬁed, we call 푇 a twin tree, or a twin pair of trees, or a twinning of 푇 +, 푇 −. We say two vertices 푥, 푦 ∈ 푇 are opposite if 훿*(푥, 푦) = 0. We call edges 푒+ ∈ 퐸푇 + and 푒− ∈ 퐸푇 − opposite if their vertices are opposites. We will sometimes denote the opposition relation with opp.

Remark 1.4.2. One motivation for the above definition is to generalize the notion of being “opposite” in finite trees; if a tree 푋 is finite, then there is always a longest geodesic path in 푋. This allows one to consider two vertices opposite if they lie at extremes of a longest geodesic. But for an infinite tree, this is no longer the case. By considering two infinite trees and a codistance function, we have a generalized way of declaring vertices “opposite.”

Remark 1.4.3. We shall employ the symbols + and − to distinguish between trees in a twinning. When mentioning elements of a twin tree 푇 = (푇 +, 푇 −, 훿*), such as vertices or edges, we shall use + and − as superscripts. If 푇 + and 푇 − are Bruhat-Tits trees arising from discrete valuations, then we have items not directly apart of the twinning itself, and for those we shall employ + and

− as subscripts to distinguish them. For example, we would write 휈+ and 휈− or 풪+ and 풪− for the discrete valuations and DVR’s, if 푇 + and 푇 − were Bruhat-Tits trees.

+ − Example 1.4.4. We can twin the tree 풞∞ with itself. Set 푇 = 풞∞, 푇 = 풞∞, and deﬁne the co-distance function by:

* 훿 (푥푛, 푦푚) = |푛 − 푚| for 푛, 푚 ∈ ℤ. Notice that each vertex in 푇 휀 has exactly one opposite in 푇 −휀.

+ 푇 푥−2 푥−1 푥0 푥1 푥2 푥3

푇 − 푦2 푦1 푦0 푦−1 푦−2 푦−3 33 A vertex can have more than one opposite, but to ensure that every vertex has at least one opposite, one must assume that the twinned trees each have vertices with valency at least two, as in our previous example. This is indeed sufﬁcient for the existence of opposites.

Lemma 1.4.5. If the trees 푇 +, 푇 − have valency at least 2, then every vertex 푥 ∈ 푇 has at least one opposite.

Proof. We show this for valency of 푇 exactly 2, from which any higher valency result follows. Let 푥 ∈ 푉 푇 . Without loss of generality, let us assume 푥 = 푥+ ∈ 푉 푇 +. Pick any 푦− ∈ 푉 푇 − and let 푚 = 훿*(푥+, 푦−). If 푚 = 0, then we are done! Else, from the (codist) condition, there exists a unique 푦′− ∈ 푉 푇 − adjacent to 푦− such that 훿*(푥+, 푦′−) = 푚 + 1. Hence, because 푇 − has valency 2, 푦− has another adjacent vertex 푧− different from 푦′−. And because 푦′− is the unique vertex adjacent to 푦− with 훿*(푥+, 푦′−) = 푚 + 1, we must have 훿*(푥+, 푧−) = 푚 − 1. So via induction on 푚, we obtain an opposite for 푥+.

Remark 1.4.6. Note from the above lemma and the deﬁnition that 훿*(푥+, 푦−) is the shortest distance (in 푇 −) from 푦− to an opposite of 푥+. By symmetry, the codistance is also the shortest distance in 푇 + from 푥+ to an opposite of 푦−.

Recall in a tree 푇 , two vertices 푥 and 푦 are said to have the same type if and only if 훿(푥, 푦) is even Deﬁnition 1.1.16. We have the exact same deﬁnition for twin trees, except using co-distance in place of distance:

Deﬁnition 1.4.7. Let (푇 +, 푇 −, 훿*) be a twin tree and let 푥+ ∈ 푉 푇 +, 푦− ∈ 푉 푇 −. We say 푥+ and 푦− are of the same type if and only if 훿*(푥+, 푦−) is even.

Proposition 1.4.8. Type is an equivalence relation on 푉 푇 + ⋃︀ 푉 푇 −, and there are only two equivalence classes: 0 and 1 (for even and odd).

Proof. To say two vertices 푥, 푦 ∈ 푉 푇 + ⋃︀ 푉 푇 − are of the same type means one of 훿+(푥, 푦), 훿−(푥, 푦), or 훿*(푥, 푦) is even (dependent upon where 푥 and 푦 live). Let us show type has the reﬂexive, symmetric, and transitive properties. 34 Reﬂexive: Let 푥 ∈ 푉 푇 + ⋃︀ 푉 푇 −. Then clearly, 훿+(푥, 푥) = 0 or 훿−(푥, 푥) = 0. So 푥 is of the same type as itself.

Symmetric: Suppose 푥, 푦 ∈ 푉 푇 + ⋃︀ 푉 푇 − are of the same type. By deﬁnition, 훿± and 훿* are symmetric. So if 훿±,*(푥, 푦) is even, then 훿±,*(푦, 푥) = 훿±,*(푥, 푦) is even.

Transitive: Suppose 푥, 푦, 푧 ∈ 푉 푇 + ⋃︀ 푉 푇 − with 푥 and 푦 of the same type and 푦 and 푧 of the same type. There are two cases: all three vertices lie in the same tree; or two lie in one tree and the third lies in the other. If all three vertices lie in the same tree, then we have a geodesic [푥, 푦] of even length from 푥 to 푦, and another geodesic [푦, 푧] of even length from 푦 to 푧. If the intersection [푥, 푦] ∩ [푦, 푧] = ∅, then by joining these paths, we will have an even length geodesic from 푥 to 푧. Else, the joining of paths [푥, 푦] and [푦, 푧] will have a backtracking. Denote the lengths of [푥, 푦], [푦, 푧], and [푥, 푦] ∩ [푦, 푧] by 푛, 푚, and 푘. Then the length of the reduced joined path is 푛 − 푚 − 2푘, which will be even. Hence 푥 and 푧 have the same type. The following argument takes care of the other case. Suppose 푥+, 푦+ ∈ 푉 푇 + have the same type. Then 훿+(푥+, 푦+) is even. Let 푧− ∈ 푉 푇 − and consider 훿*(푥+, 푧−). We have a geodesic of even length between 푥+ and 푦+, so by the deﬁnition of co-distance, as we move between adjacent vertices along this geodesic in 푇 +, we will change the co-distance with 푧− by ±1. Since 푥+ and 푦+ are an even distance apart, the parity of 훿*(푥+, 푧−) is unchanged for 훿*(푦+, 푧−). Thus, 훿*(푥+, 푧−) is even (reps. odd) if and only if 훿*(푦+, 푧−) is even (reps. odd).

Deﬁnition 1.4.9. A twin tree 푇 = (푇 +, 푇 −, 훿*) is called thick if every vertex in either tree has

휀 휀 휀 valency at least 3. In other words, |St푇 휀 (푣 )| ≥ 3 for all 푣 ∈ 푉 푇 , for 휀 = + and 휀 = −.

Proposition 1.4.10 ([RT94, Proposition 1]). Let (푇 +, 푇 −, 훿*) be a thick twin tree. Then vertices of the same type have the same valency, and the twinned trees 푇 + and 푇 − are isomorphic.

Proof. Consider two opposite vertices 푥+ and 푥− (we know opposites exist because the trees are thick). Let 푦− be adjacent to 푥−. Then since 훿*(푥+, 푥−) = 0, we have 훿*(푥+, 푦−) = 1 (because 35 co-distance is always non-negative and changes by ±1 when we move to an adjacent vertex). By deﬁnition of a twin tree, there exists a unique vertex 푦+ adjacent to 푥+ such that 훿*(푦+, 푦−) = 2. Because the above is clearly symmetric between + and −, we have that given two opposite vertices 푥+ and 푥−, there is a bijection between the neighbors of 푥+ and the neighbors of 푥− (a neighbor is an adjacent vertex). Hence, opposite vertices must have the same valency. Now suppose 푥+, 푦+ ∈ 푉 푇 + have the same type. Without loss of generality, assume that 훿+(푥+, 푦+) = 2. Because 푇 + is a tree, there is a unique vertex, say 푧+, adjacent to both 푥+ and 푦+. Because our trees are thick, 푧+ has at least one opposite. Let 푧− be opposite to 푧+. By the above bijection, there are unique vertices 푥− and 푦− adjacent to 푧− such that

훿*(푥+, 푥−) = 2 and 훿*(푦+, 푦−) = 2.

Because 푇 − is thick, there must exist a vertex adjacent to 푧−, say 푤−, such that 푤− ̸= 푥− ̸= 푦−. Furthermore, since 훿*(푧+, 푧−) = 0, then 훿*(푧+, 푤−) = 1. Now by the uniqueness of 푥− and 푦−, we must have 훿*(푥+, 푤−) = 0 and 훿*(푦+, 푤−) = 0; i.e. 푥+ opp 푤− and 푦+ opp 푤−. By the above, we know opposite vertices have the same valency. In particular, we have a bijection between the neighbors of 푥+ and the neighbors of 푤−. And we have a bijection between the neighbors of 푦+ and the neighbors of 푤−. So 푥+ and 푦+ must have the same valency. The above argument is clearly symmetric between + and −, and extends easily to vertices any even distance apart. Thus, vertices of the same type must have the same valency. Since there are only two types of vertices, there are only two valency sizes. Clearly, up to isomorphism, there is only one tree with two given valencies. Hence, 푇 + and 푇 − are isomorphic.

We see from the above theorem that, if 푇 = (푇 +, 푇 −, 훿*) is a thick twin tree, then

+ ∼ − ∼ 푇 = 푇 = 푇푘,푙, where 푇푘,푙 is a tree with valency 푘 for even type and valency 푙 for odd type. Trees such as this are said to be semi-homogeneous.

Example 1.4.11. Below is (part of) a 푇3,4 tree. Any tree twinned with 푇3,4 will be isomorphic to

푇3,4. 36

Figure 1.1: Part of a tree with two valency types: 푇3,4

Example of a Twin Tree

We now present an example of a twin tree. Let 퐾 = 푘(푡). Recall 휈+, the valuation at 0

∑︀푁 푖 (Example 1.3.4) and 휈−, the valuation at ∞ (Example 1.3.5). For 푖=푛 푎푖푡 in 퐾 = 푘(푡), the valuations are given by:

(︃ 푁 )︃ (︃ 푁 )︃ ∑︁ 푖 ∑︁ 푖 휈+ 푎푖푡 = 푛, 휈− 푎푖푡 = 푁 푖=푛 푖=푛

Or put another way: (︂ )︂ (︂ 푛 )︂ 푓(푡) 푡 푓0(푡) 휈+ = 휈+ 푚 = 푛 − 푚 푔(푡) 푡 푔0(푡) (︂푓(푡))︂ 휈 = deg(푔) − deg(푓) − 푔(푡)

where 푓0(0) ̸= 푔0(0) ̸= 0. Let 푉 be a two-dimensional vector space over 퐾 with basis {푒1, 푒2}. We

+ − let 푇 denote the Bruhat-Tits tree of 푉 with respect to 휈+; likewise, let 푇 be the Bruhat-Tits tree

of 푉 with respect to 휈− (Section 1.3). We have valuations rings 풪+ and 풪− having uniformizer elements 푡 and 푡−1, respectively. Set 퐴 = 푘[푡, 푡−1], the ring of Laurent polynomials. Note that

−1 풪+ ∩ 퐴 = 푘[푡] and 풪− ∩ 퐴 = 푘[푡 ]. To construct Bruhat-Tits trees, we had to begin with a vector space and look at lattices. To 37 construct a twinning of 푇 + and 푇 −, we require a similar starting point. Let 푀 be a free 퐴-module

+ − spanned by a basis of 푉 (say maybe 푀 contains {푒1, 푒2}). Let 퐿 , 퐿 be arbitrary 풪+, 풪− lattices, respectively, in 푉 that correspond to vertices 푥+, 푦− in our two trees. Then we set

+ − 푋+ = 퐿 ∩ 푀, 푌− = 퐿 ∩ 푀.

−1 Note that, by ??, 푋+ is a 푘[푡]-module and 푌− is a 푘[푡 ]-module. Consider the set

푛 {푡 푋+ ∩ 푌− | 푛 ∈ ℤ}

+ − 푛 Because 퐿 and 퐿 are ﬁnite dimensional over 푘, so are 푋+ and 푌−; hence so is each 푡 푋+ ∩ 푌−.

푛 Thus, we have for some integer 푁, 푡 푋+ ∩ 푌− = 0 if and only if 푛 > 푁; and, there exists an

푛 integer 푁0 such that 푛 ≤ 푁0 if and only if 푡 푋+ ∩ 푌− contains a basis for 푉 . Using these integers

푁 and 푁0, we can deﬁne a co-distance:

* + − 훿 (푥 , 푦 ) = 푁 − 푁0

Lemma 1.4.12. The co-distance function deﬁned above is well-deﬁned, and invariant under change of + and −.

+ 푛 Proof. Without loss, we can suppose 퐿 has basis {푡 푒1, 푒2}. If we take a lattice different from

+ + 푐 + 푐 + 푛+푐 푐 푛 퐿 lying in 푥 , say 푡 퐿 , then 푡 퐿 = ⟨푡 푒1, 푡 푒2⟩. Thus, we will get that 푡 푋+ ∩ 푌− = 0 if and

푛 only if 푛 > 푁 − 푐; and 푡 푋+ ∩ 푌− will contain a basis for 푉 if and only if 푛 ≤ 푁0 − 푐. Then

* + − (푁 − 푐) − (푁0 − 푐) = 푁 − 푁0 = 훿 (푥 , 푦 )

Hence the integer valued co-distance will be the same using a different, but equivalent, lattice 푡푐퐿+.

푛 −푛 Lastly, we clearly have an isomorphism between 푡 푋+ ∩ 푌− and 푋+ ∩ 푡 푌−, where the map is just multiplication by 푡−푛. 38 To summarize a little, suppose we have vertices 푥+ ∈ 푉 푇 +, 푦− ∈ 푉 푇 −. Now 퐿+ ∈ 푥+ and

− − * + − 퐿 ∈ 푦 can be chosen so that 푁0 = 0, making 훿 (푥 , 푦 ) = 푁. This says that 푋+ ∩ 푌− contains

a basis for 푉 , but 푡푋+ ∩ 푌− does not.

* + + − − If we have 훿 (푥+, 푦−) = 0, then one can ﬁnd lattices 퐿 ∈ 푥 , 퐿 ∈ 푦 such that, if {푒1, 푒2} is a basis for 푋+ ∩ 푌−, then 푋+ = ⟨푒1, 푒2⟩푘[푡] and 푌− = ⟨푒1, 푒2⟩푘[푡−1].

* + − 푁 If 훿 (푥 , 푦 ) = 푁 > 0, then 푋+ = ⟨푒1, 푒2⟩푘[푡] and 푌− = ⟨푡 푒1, 푒2⟩푘[푡−1]. Of course, one should actually prove that the co-distance function deﬁned above actually gives

+ a twinning of 푇 and 푇−. This is done in [RT94]:

* + − Lemma 1.4.13. [RT94, Lemma 2.1] Using the notations above, we have 훿 (푥 , 푦 ) = dim푘(푡푋+∩

푌−) = dim푘(푋+ ∩ 푌−) − 2.

Proposition 1.4.14 ([RT94, Proposition 2.2]). The co-distance function described above gives a twinning of 푇+ and 푇−.

More examples of twin trees are discussed at the end of [AR09]. One variation of the above example is to consider the completed valuations 휈̂︁+ and 휈̂︁− as discussed in Example 1.3.23. Precisely, we can twin, via the same manner as above, the trees 푋+ and 푋−, where 푋+ is the Bruhat-Tits

− −1 tree for 푘((푡)), the completion of 퐾 with respect to 휈+; and 푋 is the Bruhat-Tits tree for 푘((푡 )), the completion of 퐾 with respect to 휈−. See [AR09, Example 66]. However, we note that the trees 푇 ± and 푋± are isomorphic. Thus, one can assume, for convenience, that the underlying ﬁeld is necessarily complete.

Twin Apartments

In a tree 푇 , an apartment is a path without repeated edges or end-points. So an apartment is a bi-infinite line. A half-apartment is a path with only one end-point, hence is a 푇 -ray as defined in Definition 1.1.16. Thus, each apartment has two corresponding ends. If 휀 and 휂 are two ends, then these ends determine a unique apartment. Apartments are important components of trees, as we can think of a tree as a collection of apartments that are “glued” together nicely (indeed, this is how one would define a tree as a building 39 [AB08, 4.1]).

+ Example 1.4.15. Consider 푇 , the Bruhat-Tits tree for the valuation 휈+ on 퐾 = 푘(푡). For each

+ 푛 integer 푛, let 퐿푛 be the 풪+-lattice with basis {푡 푒1, 푒2}. Then the corresponding lattice classes

+ 푛 + + 푥푛 = [⟨푡 푒1, 푒2⟩풪+ ] form an bi-inﬁnite path in 푇 . The vertices 푥푛 determine this bi-inﬁnite path,

+ + + hence we write Σ = {푥푛 | 푛 ∈ ℤ} and use this as our notation for an apartment in 푇 .

− − Similarly, for the tree 푇 arising from the valuation at ∞ on 푘(푡), let 퐿푛 be the 풪−-lattice with

푛 − − 푛 −푛 basis {푡 푒1, 푒2}. The corresponding vertices in 푇 are 푥푛 = [⟨푡 푒1, 푒2⟩풪− ] = [⟨푒1, 푡 푒2⟩풪− ] and

− − − these form an apartment Σ = {푥푛 | 푛 ∈ ℤ} in 푇 . As an example of the co-distance function, we have

* + − 훿 (푥푛 , 푥푛 ) = 0

+ + * + − for all 푛 ∈ ℤ. If we ﬁx 푥 = 푥0 , and set 푠푛 = 훿 (푥 , 푥푛 ), then we have 푠푛 = 푛 for all 푛.

+ * If we have a twin pair of trees 푇 = (푇 , 푇−, 훿 ), then we would like a “twinned” version of apartments. We deﬁne these as follows:

+ * + + − Deﬁnition 1.4.16. Let 푇 = (푇 , 푇−, 훿 ) be a twin tree, Σ an apartment in 푇 , and Σ an apartment in 푇 −. Then we call Σ = (Σ+, Σ−) a twin-apartment or twin pair of apartments if, given any vertex 푣 ∈ Σ휀, there exists a unique vertex 푤 ∈ Σ−휀 opposite 푣 (휀 ∈ {+, −}).

Consider a pair of non-opposite vertices 푥+ and 푦− in a twin tree 푇 = (푇 +, 푇 −, 훿*). Because 푥+ and 푦− are non-opposite, they have 훿*(푥+, 푦−) > 0. Hence, by the (Codist) property, there

+ + + + * + − exists a unique ray 훾푥+ = (푥 , 푥1 , 푥2 ,...) starting at 푥 such that 훿 (푥푖 , 푦 ) increases as 푖 → ∞.

− − − + Likewise, there us a unique ray 훾푦− = (푦 , 푦1 , 푦2 ,...) such that the co-distance from 푥 increases along the path (cf. [RT94, prop. 3.1-3.4]).

+ + − − Now, the rays 훾푥+ and 훾푦− correspond to ends 푒 in 푇 and 푒 in 푇 (Definition 1.1.16). According to [RT94, Prop 3.4], the ends 푒+ and 푒− can be “identified”. Hence by considering all pairs of non-opposite vertices, we can consider the collection of all such identified pairs of ends, called the ends of the twinning or ends of a twin pair of trees. 40 Now we state two important results concerning twin apartments:

Proposition 1.4.17 ([RT94, Prop 3.5]). Every pair of opposite edges is contained in a unique twin apartment Σ = (Σ+, Σ−), and the apartments Σ+ and Σ− have the same two ends.

Proposition 1.4.18 ([RT94, Prop 3.6]). In a thick pair of twin trees, the set of twin apartments uniquely determines the twinning.

+ − * Example 1.4.19. Let 푇 = (푇 , 푇 , 훿 ) be the twin tree given above in Section 1.4. Let {푒1, 푒2} be a basis for the 푘[푡, 푡−1]-module 푀. Let Σ+ and Σ− be the apartments deﬁned in Example 1.4.15 for the twin tree 푇 = (푇 +, 푇 −, 훿*) associated to 푘[푡, 푡−1]. Then Σ = (Σ+, Σ−) is a twin apartment.

휀 휀 −휀 −휀 If 푥푛 is a vertex in Σ , then its unique opposite vertex in Σ is 푥푛 .

The two ends of the twin apartment Σ are given by the 1-spaces ⟨푒1⟩ and ⟨푒2⟩ in 푉 . If we read off the vertices from the end ⟨푒1⟩ to the end ⟨푒2⟩, then we have

+ + + + + + Σ+ 푥−2 푥−1 푥0 푥1 푥2 푥3

Σ− − − − − − − 푥2 푥1 푥0 푥−1 푥−2 푥−3

All the twin apartments for 푇 are in fact given by bases for 푀, so a change in choice of basis for 푀 will result in a different twin apartment (cf. [AB08, 6.12]).

Automorphisms of Twin Trees

Now let T = (푇 +, 푇 −, 훿*) be a twin tree. We deﬁne an automorphism of a twin tree as a pair of automorphisms from 푇 + and 푇 − which preserve the co-distance function:

Deﬁnition 1.4.20 ([RT94], [AR09, Def. 58]). An automorphism of the twin tree T = (푇 +, 푇 −, 훿*)

+ − is a pair (훼+, 훼−) of automorphisms of 푇 , respectively 푇 , such that

* + − * + − 훿 (훼+(푥 ), 훼−(푦 )) = 훿 (푥 , 푦 )

for any 푥+ ∈ 푉 푇 + and 푦− ∈ 푉 푇 −. 41 Let 퐴 = Aut(T ) denote the automorphism group for a twin tree T = (푇 +, 푇 −, 훿*). Recall from Proposition 1.4.10 that, if T is a thick twin tree, then the twinned trees 푇 + and 푇 − are semi- homogeneous. Moreover, there are only two types of vertices in T (Proposition 1.4.8). We do not require automorphisms of T to be type-preserving. We denote the subgroup of type preserving automorphisms of 퐴 by 퐴0. Clearly we have that [퐴 : 퐴0] ≤ 2. Similarly, we can deﬁne an isomorphism of two twin trees:

+ − * + − * Deﬁnition 1.4.21. Let T0 = (푇0 , 푇0 , 훿0) and T1 = (푇1 , 푇1 , 훿1) be two twin trees. An isomor-

+ + − − phism 휙 : T0 → T1 is given by a pair 휙 = (휙+, 휙−) with 휙+ : 푇0 → 푇1 and 휙− : 푇0 → 푇1 tree isomorphisms, satisfying

* + − * + − 훿1(휙+(푥 ), 휙−(푦 )) = 훿0(푥 , 푦 )

+ + − − for all 푥 ∈ 푉 푇0 , 푦 ∈ 푉 푇0 .

Now assume 푇 is a thick tree. In [RT99] (the epic sequel to [RT94]) Ronan and Tits develop the notion of a universal twin 푇 *, which contains all possible twinnings of 푇 . They in fact show that, to any semi-homogeneous tree, there are a large number of non-isomorphic twinnings. Precisely,

Theorem 1.4.22 ([RT99, Th. 8.2]). Let 푇 be a thick, semi-homogeneous tree with |푉 푇 | = 훼 (the cardinality of the tree 푇 ). Then 푇 belongs to 2훼 isomorphism classes of twinnings, and among these there are 2훼 having trivial automorphism group. 42

CHAPTER 2 GROUPS ACTING ON TREES - BASS-SERRE THEORY

In this chapter we present the basis of Bass-Serre theory, which gives a complete characterization of groups that act on trees. The material in this chapter is a prerequisite for Chapter 3 where we discuss Tree Lattices. Section 2.5 contains an application in the situation of twin-trees, and is an example of a Nagao Lattice, which we discuss in Section 3.8. More applications will be discussed in Chapter 4.

2.1 Free Products, Amalgams and HNN Extensions

In this section, we recall various constructions of free products of groups. In particular, amalgamated free products and HNN extensions. These are the principal group constructions of Bass-Serre theory. Our references are [Bog08] and [Ser80].

Free Products

Recall that free groups are groups with no relations. Any free group has a basis - a linearly ordered subset for which any nontrivial element of the group can be uniquely represented by a product of elements from the basis. The additive group of integers ℤ is an example of a free group, with a basis consisting of one element. A free product is an operation taking as input two groups and producing a new group.

Deﬁnition 2.1.1. Suppose 퐴 and 퐵 are groups. We denote by 퐴 * 퐵 the free product of the groups 퐴 and 퐵. The elements of 퐴 * 퐵 are all reduced words:

푔1푔2푔3 ··· 푔푘−1푔푘

where 푘 ≥ 0, 푔푖 ∈ (퐴 ∪ 퐵) ∖ {1}, 1 ≤ 푖 ≤ 푘, and adjacent letters 푔푖, 푔푖+1 do not lie in the same group 퐴 or 퐵. The identity element is represented by the empty word. The multiplication in 퐴 * 퐵 is given by inductive juxtaposition to ensure the product is a reduced word. Precisely, if 43

푥 = 푔1푔2 ··· 푔푘 and 푦 = 푕1푕2 ··· 푕푚 are nontrivial reduced words in 퐴 * 퐵, then

⎧ ⎪ ⎪푔1 ··· 푔푘푕1 ··· 푕푚 if 푔푘 and 푕1 lie in different groups, ⎪ ⎨⎪ 푥 · 푦 = 푔1 ··· 푔푘−1푧푕2 ··· 푕푚 if 푔푘, 푕1 lie in the same group and 푧 = 푔푘푕1 ̸= 1, ⎪ ⎪ ⎪ ⎩⎪(푔1 ··· 푔푘−1) · (푕2 ··· 푕푚) otherwise.

The notion that the above product is free is best seen in the following way:

Theorem 2.1.2 ([Bog08, Ch. 2, 10.3]). Suppose we have groups presentations 퐴 = ⟨푋 | 푆⟩ and 퐵 = ⟨푌 | 푅⟩ and 푋 ∩ 푌 = ∅. Then

퐴 * 퐵 = ⟨푋 ∪ 푌 | 푆 ∪ 푅⟩ .

Thus, a free product of two groups is obtain by simply combining generators and relations. However, a free product is not necessarily a free group; if either of 푅 or 푆 is non-empty, then 퐴 * 퐵 is not a free group.

Example 2.1.3. Let 퐴 = ⟨푎⟩ and 퐵 = ⟨푏⟩; so both groups are inﬁnite cyclic groups. In particular, ∼ ∼ ∼ 퐴 = 퐵 = ℤ. Then 퐴 * 퐵 = ⟨푎, 푏⟩ = ℤ * ℤ is a free group with basis set {푎, 푏}. A few elements of this group: 푎푏푎푏푎푏, 푎2푏3푎4푏5, etc.

Example 2.1.4. Recall that the inﬁnite dihedral group 퐷∞ is the automorphism group of the tree ∼ 풞∞ (Example 1.1.9). It can be shown [Bog08, Ch.2, 10.4] that 퐷∞ = 푍2 * 푍2, the free product of two cyclic groups of order two. Speciﬁcally, the automorphisms 푎 and 푐 = 푏푎 are both of order two and generate the group:

⟨︀ 2 2 ⟩︀ 퐷∞ = 푎, 푐 | 푎 = 푐 = 1 .

Geometrically, one can see the generators 푎 and 푐 as the automorphisms that reﬂect the graph 풞∞

through the point 0 and the middle of the edge 푒0, respectively.

Clearly, 퐷∞ is not a free group, hence this is an example of a free product not being a free group. 44 One can generalize a free product between two groups to an arbitrary collection of groups.

Deﬁnition 2.1.5. Let {퐺훼}훼∈퐼 be a collection of groups. Then *훼∈퐼 퐺훼 is a group, called the free product of the groups 퐺훼, consisting of all reduced words:

푔1푔2푔3 ··· 푔푘−1푔푘

where 푘 ≥ 0, 푔푖 ∈ 퐺훼푖 ∖ {1} for some 훼푖 ∈ 퐼, 1 ≤ 푖 ≤ 푘, and adjacent letters 푔푖, 푔푖+1 do not lie in

the same group 퐺훼푖 . The product between two elements 푥 and 푦 is deﬁned as in Deﬁnition 2.1.1.

Amalgamated Free Products

Suppose we have two groups 퐺 and 퐻, each containing an isomorphic copy of another group 퐴. Then one can form a quotient group from the free product 퐺 * 퐻 by modding out by the shared subgroup 퐴.

Deﬁnition 2.1.6. Let 퐺 and 퐻 be groups. Suppose 퐺 and 퐻 contain isomorphic subgroups 퐴 and 퐵. We ﬁx an isomorphism 휙 : 퐴 → 퐵. The amalgamated free product of 퐺 and 퐻 by the subgroups 퐴 and 퐵 is the quotient (퐺 * 퐻)/푁, where

푁 = ⟨︀⟨︀휙(푎)푎−1 | 푎 ∈ 퐴⟩︀⟩︀ ,

−1 the normal closure of the set {휙(푎)푎 | 푎 ∈ 퐴}. We denote this product in several ways: 퐺 *퐴 퐻,

퐺 *퐴=퐵 퐻, 퐺 *휙 퐻. We shall also refer to this product by the following presentation:

⟨퐺 * 퐻 | 푎 = 휙(푎), 푎 ∈ 퐴⟩ .

2 Example 2.1.7. Let 퐺 = 푍4 = ⟨푥⟩ and 퐻 = 푍6 = ⟨푦⟩. Then the subgroups 퐴 = ⟨푥 ⟩ ≤ 퐺 and 45 퐵 = ⟨푦3⟩ ≤ 퐻 are isomorphic:

휙 : 퐴 → 퐵

푥2 ↦→ 푦3

∼ ∼ and clearly 퐴 = 퐵 = 푍2. Then the amalgamated free product can be written via generators and relations as:

⟨︀ 4 6 2 3⟩︀ 푍4 *푍2 푍6 = 푥, 푦 | 푥 = 푦 = 1, 푥 = 푦 .

3 So for example, consider the reduced word 푤 = 푥푦 푥 in the free product 푍4 * 푍6. In the free

2 3 product, 푤 is non-trivial. However, in the quotient group 푍4 *푍2 푍6, we obtain the relation 푥 = 푦 and hence 푤 reduces to the identity element in the amalgamated product:

푤 = 푥푦3푥 = 푥푥2푥 = 푥4 = 1.

As with free products, we can generalize an amalgamted free product beyond two groups to an arbitrary collection {퐺훼}훼∈퐼 , where each group 퐺훼 contains an isomorphic copy of a ﬁxed group

퐴. In other words, we require for each 훼 ∈ 퐼, an injective homomorphism 휙훼 : 퐴 → 퐺훼, and we identify 퐴 with its image in each of the 퐺훼. Then we form a quotient as before,

−1 (*훼퐺훼) /⟨⟨휙훼(푎)푎 | 푎 ∈ 퐴, 훼 ∈ 퐼⟩⟩

and call this the sum of the 퐺훼 with 퐴 amalgamated. We denote this group by *퐴퐺훼. Note that this group can also be seen as the direct limit of the family (퐴, 퐺훼, 훼 ∈ 퐼) with respect to the homomorphisms 휙훼 [Ser80, I.1.2]. An even more general notion of amalgam is deﬁned in [Ser80, I.1.1].

Example 2.1.8. Given a ﬁnite collection of groups, we can easily form an amalgam sum by per- forming successive amalgams. For instance, if we have groups 퐺1, 퐺2, 퐺3, each with isomorphic 46 copies of a ﬁxed group 퐴, then we can form the amalgam 퐺1 *퐴 퐺2 *퐴 퐺3 as (퐺1 *퐴 퐺2) *퐴 퐺3.

For instance, let 퐺1 = 푍4 = ⟨푥⟩, 퐺2 = 푍6 = ⟨푦⟩, 퐺3 = 푍12 = ⟨푧⟩ and 퐴 = 푍2 = ⟨푎⟩. The

2 3 6 three injective morphisms 퐴 → 퐺푖, 푖 = 1, 2, 3 are, respectively, 푎 ↦→ 푥 , 푎 ↦→ 푦 , and 푎 ↦→ 푧 .

We formed 퐺1 *퐴 퐺2 in Example 2.1.7. If we take that result and amalgamate with 퐺3, we obtain the following:

4 6 1 2 3 6 푍4 *푍2 푍6 *푍2 푍12 = ⟨푥, 푦, 푧 | 푥 = 푦 = 푧 2, 푥 = 푦 = 푧 ⟩.

As mentioned above, there is a more general notion of amalgam in [Ser80, I.1.1]. We will not

utilize this completely, but only in the following way [Ser80, I.1.2, p.4-5]: suppose {퐺훼}훼∈퐼 is a family of groups, 퐷 a set of pairs of distinct elements of 퐼, and for all {훼, 훽} ∈ 퐷 we have a

′ group 퐴훼훽 and injective homomorphisms 휙훼훽 : 퐴훼훽 → 퐺훼, 휙훼훽 : 퐴훼훽 → 퐺훽. Then we can form a group 퐺 like before out of the groups 퐺훼,

퐺 = lim(퐺훼, 퐴훼훽). −→

The simplest description of 퐺 is via generators and relations. The generating set for 퐺 is given by the disjoint union of the generating sets of the 퐺훼; the relations are those already present for each

−1 ′ −1 퐺훼, and 휙훼훽(푎)푎 = 1 and 휙훼훽(푎)푎 = 1, where 푎 ∈ 퐴훼훽, for all pairs {훼, 훽} ∈ 퐷.

Example 2.1.9. We again consider three cyclic groups 퐺1 = 푍4 = ⟨푥⟩, 퐺2 = 푍6 = ⟨푦⟩, and

퐺3 = 푍12 = ⟨푧⟩. But instead of a single ﬁxed group like in Example 2.1.8, we will have two: 47

퐴 = 푍2 = ⟨푎⟩ and 퐵 = 푍3 = ⟨푏⟩. We have the following injective homomorphisms:

휙12 : 퐴 → 퐺1

푎 ↦→ 푥2

′ 휙12 : 퐴 → 퐺2

푎 ↦→ 푦3

휙23 : 퐵 → 퐺2

푏 ↦→ 푦2

′ 휙23 : 퐵 → 퐺3

푏 ↦→ 푧4

Thus our set of pairs 퐷 = {(1, 2), (2, 3)}, and we could just as well write 퐴12 = 퐴 and 퐴23 = 퐵. The the amalgamated sum of these three groups with respect to the above homomorphisms is:

4 6 1 2 3 2 4 푍4 *푍2 푍6 *푍3 푍12 = ⟨푥, 푦, 푧 | 푥 = 푦 = 푧 2 = 1, 푥 = 푦 , 푦 = 푧 ⟩.

Note that, like in Example 2.1.8, we could perform successive amalgams and obtain the same result.

Example 2.1.10. Here we give one more example, but with an inﬁnite family of groups. This example will in fact be applied later in this chapter (cf. Theorem 2.5.1).

Let {퐺푖}푖≥1 be a collection of groups such that 퐺푖 ≤ 퐺푖+1 for all 푖 ≥ 1. Our set 퐷 will consist of all consecutive pairs of positive integers 퐷 = {(푖, 푖+1) | 푖 = 1, 2, 3,... }. For each pair (푖, 푖+1)

′ we set 퐴푖,푖+1 = 퐺푖. Then the injective morphisms 휙푖,푖+1 : 퐺푖 → 퐺푖 and 휙푖,푖+1 : 퐺푖 → 퐺푖+1 are the identity map and inclusion, respectively.

We amalgamate the groups 퐺푖 inductively. Set 퐻푖 = 퐺푖 *퐺푖 퐺푖+1 for 푖 = 1, 2, 3,.... Since

퐺푖 ≤ 퐺푖+1, this amalgam “collapses” into just 퐺푖+1. Hence, for each 푖, 퐻푖 = 퐺푖+1. Then it 48 follows that the amalgamated product of the groups 퐺푖 is simply the union of all the 퐺푖:

⋃︁ 퐻 = lim퐻푖 = 퐺푖. −→ 푖≥1

HNN Extensions

We have one more group construction that is fundamental to Bass-Serre theory. The following is called an HNN extensions after the mathematicians G. Higman, B.H. Neumann, and H. Neumann [HNN49].

Deﬁnition 2.1.11. Let 퐺 be a group and 퐴 and 퐵 isomorphic subgroups of 퐺. Let 휃 : 퐴 → 퐵 be an isomorphism. Let 푡 be a new element not in 퐺 and 푆 = ⟨푡⟩ the inﬁnite cyclic group generated by 푡. ∼ Then the HNN extension of 퐺 derived from 퐴 = 퐵 and 휃 is the quotient group 퐺*퐴 = (퐺 * 푆) /푁, where 푁 = ⟨⟨푡−1푎푡(휃(푎))−1 | 푎 ∈ 퐴⟩⟩.

The element 푡 is called the base letter and the group 퐺 is the base of 퐺*퐴. We will write 퐺*퐴 with the following notation:

−1 퐺*퐴 = ⟨퐺, 푡 | 푡 푎푡 = 휃(푎), 푎 ∈ 퐴⟩.

Example 2.1.12. The integers can be seen as a trivial example of an HNN extension:

ℤ = {1}*{1} .

So as in the deﬁnition, 퐺 = {1}, the trivial group and the associated groups are 퐴 ∼= 퐵 = {1} with 휃 a trivial isomorphism. Hence,

−1 퐺*퐴 = ⟨퐺, 푡 | 푡 푎푡 = 휃(푎), 푎 ∈ 퐴⟩

= ⟨푡 | 푡−1푡 = 1⟩ ∼ = ⟨푡⟩ = ℤ. 49

One can show that any element in 퐺*퐴 can be written uniquely in a normal form and 퐺 and

푆 can be embedded into 퐺*퐴. Moreover, the subgroups 퐴 and 퐵 will be conjugate in 퐺*퐴 (after identifying 퐺 and 푆 with their images in 퐺*퐴). See [Bog08, Ch.2, 14.7]. Amalgamated free products and HNN extensions have a close connection. Given 퐴 ≤ 퐺 and an injective homomorphism

−1 휙 : 퐴 → 퐺, one can form the HNN extension 퐺*퐴 = ⟨퐺, 푡 | 푡 푎푡 = 휙(푎), 푎 ∈ 퐴⟩ by means of a certain amalgamated product 퐻 (formed in a very particular way) and a semi-direct product with ∼ 푆 = ⟨푠⟩; that is, 퐺*퐴 = 퐻 ⋊ 푆. See [Ser80, I.1.4] and [Bog08, Ch. 2, 17] for precise details.

Example 2.1.13. Although we just saw that HNN extensions can be seen as semi-direct products involving amalgamated free products, nice explicit examples are given by Baumslag-Solitar groups [BS62]. Let 푝, 푞 ≥ 1 be integers (but not both equal to 1). Then we deﬁne a group by generators and relations: 퐵푆(푝, 푞) = ⟨푎, 푏 | 푎푏푝푎−1 = 푏푞⟩.

Any Baumslag-Solitar group can be seen to be an HNN extension of the integers. Let 퐺 = ℤ = ⟨푏⟩ and let 퐴 = ⟨푏푝⟩ and 퐵 = ⟨푏푞⟩. Thus 퐴 and 퐵 are subgroups of ℤ of index 푝 and 푞, respectively. Let 휃 : 퐴 → 퐵 be the isomorphism deﬁned by 휃(푏푝) = 푏푞. Then clearly from the deﬁnition above

퐵푆(푝, 푞) = 퐺*퐴 = ℤ*ℤ, the HNN extension of ℤ derived from ℤ and 휃.

2.2 Groups Acting on Graphs and Bass-Serre Trees

In the previous section, we deﬁned the quintessential group objects in Bass-Serre theory: amalgamated products and HNN extensions. In this section, we shall exhibit the close connection between these two objects and groups acting on graphs and trees. This will serve as a nice “warm- up” before the next section, where we discuss graphs of groups and their associated fundamental groups, which generalizes both amalgamated free products and HNN extensions. But this generalization will be better understood after a few more basic examples, which we discuss in this section below. This section will be light on details so as to not slow down the momentum of our exposition. However, one can refer to [Ser80] and [Bog08] where noted, for all the in-depth details. 50 In this section, assume (unless otherwise noted) that any group acting on a graph does so without inversion.

Deﬁnition 2.2.1. Let 퐺 be a group acting on a graph 푋 and let 퐺∖푋 be the corresponding quotient graph. A fundamental domain of 퐺∖푋 is a subgraph 퐹 of 푋 such that the natural projection map restricted to 퐹 , 푝 : 퐹 → 퐺∖푋 is an isomorphism.

Proposition 2.2.2 ([Ser80, I.4.1]). Let 퐺 be a group acting on a tree 푋. A fundamental domain of 퐺∖푋 exists if and only if 퐺∖푋 is a tree.

Proof. Suppose ﬁrst that 퐺∖푋 is a tree. Then by Proposition 1.2.13, there exists a subtree 푇 of 푋 such that 푝 : 푇 → 퐺∖푋 is an isomorphism. Hence, 푇 is a fundamental domain. Conversely, suppose a fundamental domain 퐹 exists for 퐺∖푋. If 퐹 is a tree, then by the isomorphism 푝 : 퐹 → 퐺∖푋, so is 퐺∖푋. Since 푋 is a tree, it is non-empty and connected, hence so is 퐺∖푋 (if 푙 is a path between vertices 푥 and 푦 in 푋, then 푝(푙) is a path between vertices 풪(푥) and 풪(푦) in 퐺∖푋). Thus, 퐹 must be non-empty and connected. Any non-empty, connected subgraph of a tree is itself a tree. Therefore, 퐺∖푋 is a tree.

The next theorem shows the connection between an amalgamated free product with two factors and group actions on trees.

Theorem 2.2.3. Suppose 퐺 = 퐺1 *퐴 퐺2, an amalgamated free product (cf. Deﬁnition 2.1.6). Then there exists a unique (up to isomorphism) tree 푋 on which 퐺 acts without inversion. The quotient graph 퐺∖푋 is a segment, a tree consisting of two vertices and two mutually inverse edges. Moreover, there exists a fundamental domain 퐹 for 퐺∖푋 푒 퐹 = 푃 푄

such that the vertex and edge stabilizers in 퐺 of 퐹 are 퐺푃 = 퐺1, 퐺푄 = 퐺2, and 퐺푒 = 퐴. 푓

Conversely, suppose we are given a group 퐻 acting on a tree 푌 and 푇 = 푅 푆 51 is a fundamental domain for 퐻∖푌 . Then the homomorphism 퐻푅 *퐻푓 퐻푆 → 퐻 induced by the

inclusions 퐻푅 → 퐻 and 퐻푆 → 퐻, is an isomorphism.

The tree 푋 is called the Bass-Serre tree associated to 퐺. The proof of Theorem 2.2.3 can be found in [Ser80, I.4.1, Thm 6, Thm 7] and [Bog08, Ch. 2, Thm 12.1, Thm 12.3]. Below we sketch

the construction of 푋 and sketch why 퐻푅 *퐻푓 퐻푆 → 퐻 is an isomorphism.

Proof. We deﬁne the tree 푋 by via left cosets of 퐺 by the groups making up its amalgam.

ᨁ 푉 푋 = (퐺/퐺1) (퐺/퐺2)

ᨁ 퐸푋 = (퐺/퐴) (퐺/퐴)

The maps 훼 : 퐸푋 → 푉 푋 and 휔 : 퐸푋 → 푉 푋 are induced by the inclusion maps 퐴 → 퐺1 and

퐴 → 퐺2. Hence, 훼(푔퐴) = 푔퐺1 and 휔(푔퐴) = 푔퐺2 for 푔 ∈ 퐺. Then we deﬁne 푔퐴 = 푔퐴, 훼(푔퐴) = 휔(푔퐴), and 휔(푔퐴) = 훼(푔퐴) making 푋 a graph. See [Bog08, 12.1] for the details on why 푋 is actually a tree. Clearly, 퐺 acts on 푋 via left multiplication. Let 퐹 be the segment in 푋 with vertices 푃 =

퐺1, 푄 = 퐺2, and positively oriented edge 푒 = 퐴, with 훼(푒) = 푃, 휔(푒) = 푄. Then from how we deﬁned 푋, we must have 풪(푃 ) ⊔ 풪(푄) = 푉 푋 and 풪(푒) ⊔ 풪(푒) = 퐸푋. Thus, 퐺∖푋 is a segment with 퐹 a fundamental domain.

For the second part of the theorem, let 휙 : 퐻푅 *퐻푓 퐻푆 → 퐻 be the homomorphism that is the

identity on 퐻푅 and 퐻푆. To see that 휙 is injective, see [Ser80, I.4.1, Lemma 3] or [Bog08, 12.3]. We will show that 휙 is surjective.

′ ′ ′ Let 퐻 = ⟨퐻푅, 퐻푆⟩ and assume that 퐻 is a proper subgroup of 퐻. Let 푌1 = 퐻 · 푇 and

′ 푌2 = (퐻 ∖ 퐻 ) · 푇 . Since 푇 is a fundamental domain for 퐻∖푌 , we have that 푌1 ∪ 푌2 = 푌 . Let

′ ′ ′ 푥 be a vertex of 푌1. If 푥 is also a vertex in 푌2, then we must have, for some 푕 ∈ 퐻 , 푕 ∈ 퐻 ∖ 퐻 either 푕′푅 = 푕푆 or 푕′푆 = 푕푅; or we have 푕′푅 = 푕푅 or 푕′푆 = 푕푆. But all of these are impossible

since 푅 and 푆 are inequivalent under the action of 퐻. Hence, it must be the case that 푌1 ∩ 푌2 = ∅. 52

But 푌 = 퐻 · 푇 is a tree, hence is connected. Thus 푌 = 푌1 ⊔ 푌2 is a contradiction. Therefore, it must be the case that 퐻′ = 퐻, showing that 휙 is surjective.

Remark 2.2.4. In the Bass-Serre tree 푋 constructed above Theorem 2.2.3, the degree of any vertex

푔퐺푖, 푖 ∈ {1, 2} is equal to the index |퐺푖 : 퐴|, the number of left cosets of 퐴 in 퐺푖. The stabilizer

−1 of 푔퐺푖 in 퐺 is 푔퐺푖푔 .

Example 2.2.5. Recall that Aut(풞∞) = 퐷∞ and 퐷∞ acts with inversion on 풞∞ (Example 1.2.5).

But on the barycentric subdivison 퐵(풞∞), we have 퐷∞ acts without inversion. We can take as fundamental domain the segment with vertices 0 and 1/2. The automorphisms 푎 and 푐 ﬁx the vertices 0 and 1/2, respectively; the edge joining 0 and 1/2 has trivial stabilizer subgroup {1}.

퐵(풞∞) −1/2 0 1/2 1

∼ ∼ Let 푆0 = ⟨푎⟩ = 푍2 and 푆1 = ⟨푐⟩ = 푍2. Below we see part of the Bass-Serre tree for ∼ 퐷∞ = 푍2 * 푍2. 푎{1} {1} 푐{1} 푐푎{1}

푎푆1 푆0 푆1 푐푆0

Example 2.2.6. We saw in the last section (Example 2.1.7) the amalgam 푍4 *푍2 푍6. In fact, this product has the following isomorphism:

∼ SL2(ℤ) = 푍4 *푍2 푍6.

Recall that SL2(ℤ) is the group of 2 × 2 matrices of determinant 1 with integer entries.

To show the above isomorphism, one must look at the natural action of SL2(ℤ) on the hyperbolic plane ℍ2. Using the upper half-plane model, we can write ℍ2 = {푧 ∈ ℤ | ℑ(푧) > 0}. Then

2 SL2(ℤ) acts by linear fractional transformations on ℍ :

⎛ ⎞ 푎 푏 푎푧 + 푏 ⎜ ⎟ ⎝ ⎠ : 푧 ↦→ 푐 푑 푐푧 + 푑 53 Let 푇 = {푒푖휃 | 휋/3 ≤ 휃 ≤ 휋/2}, a subset of ℍ2. This can be viewed as the geometric realization of a segment: 푦

푃 ↔ 푒푖휋/2 푄 ↔ 푒푖휋/3

the interior points of 푇 correspond to the edge 푦 and the endpoints 푒푖휋/2 and 푒푖휋/3 with 훼(푦) = 푃

2 and 휔(푦) = 푄, respectively. One can show that the subset 푋 = SL2(ℤ) · 푇 of ℍ is (the geometric realization of) a tree [Bog08, Ch.2, Thm 13.7].

A simple calculation shows that the stabilizers in SL2(ℤ) of the vertices 푃 and 푄 and edge 푦 are generated by the following matrices, respectively:

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 1 0 1 −1 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 퐴 = ⎝ ⎠ , 퐵 = ⎝ ⎠ , −퐼 = ⎝ ⎠ . −1 0 −1 1 0 −1

∼ ∼ ∼ That is, setting 퐺 = SL2(ℤ), we have 퐺푃 = ⟨퐴⟩ = 푍4, 퐺푄 = ⟨퐵⟩ = 푍6, and 퐺푦 = ⟨−퐼⟩ = 푍2. By the second part of Theorem 2.2.3, we obtain the isomorphism:

∼ SL2(ℤ) = 퐺푃 *퐺푦 퐺푄 = 푍4 *푍2 푍6.

Although the set 푋 is necessarily isomorphic to a semi-homogenous tree of valencies 2 and 3, the realization of this tree as a subset of ℍ2 is a nice image (cf. [DD89, p.22]) or the image below.

1 푄 푃 0.8

0.6

0.4

0.2

-2.0 -1.5 -1 -0.5 0 0.5 1 1.5 2 54

Example 2.2.7. As shown in [DD89, I.5.2], one can show that the general linear group GL2(ℤ)

2 acts on ℍ just like SL2(ℤ) (cf. Example 2.2.6). The set 푇 is also a fundamental domain for this action, and via some simple calculations, one can deduce the decomposition:

∼ GL2(ℤ) = 퐷4 *퐷2 퐷6

where 퐷푛 denotes the dihedral group of order 2푛.

2 In the same manner, the quotient groups PGL2(ℤ) and PSL2(ℤ) act on ℍ with 푇 as fundamental domain. The corresponding decompositions are:

∼ PGL2(ℤ) = 퐷2 *퐷1 퐷3,

and ∼ PSL2(ℤ) = 푍2 * 푍3.

We mention here brieﬂy another application of Theorem 2.2.3. If 휙 : 퐺 → 퐻 is an epimorphism of groups, and 퐻 has an amalgamated free product decomposition, then so does 퐺. Precisely:

Proposition 2.2.8 ([Bog08, Ch.2, 12.5]). Suppose 퐺 and 퐻 are groups and 휙 : 퐺 → 퐻 is an

−1 epimorphism. Suppose that 퐻 = 퐻1 *퐻3 퐻2. Then 퐺 = 퐺1 *퐺3 퐺2, where 퐺푖 = 휙 (퐻푖), 푖 = 1, 2, 3.

Proof. Since 퐻 = 퐻1 *퐻3 퐻2, we know by Theorem 2.2.3 that there exists a tree 푋 on which 퐻 acts without inversion and such that the quotient graph 퐻∖푋 has fundamental domain a segment 푇 : 55 푒 푇 푄 푃 ,

with 퐻푝 = 퐻1, 퐻푒 = 퐻3, and 퐻푄 = 퐻2. We deﬁne an action of 퐺 on the tree 푋 by: 푔 · 푥 := 휙(푔) · 푥, for 푔 ∈ 퐺, 푥 a vertex or edge in 푋. This is an action since 휙 is a homomorphism and the action of 퐻 on 푋 is already deﬁned. Since 퐻 acts without inversion, so does 퐺.

For any 푥 in 푋, the orbit under 퐺 is 풪퐺(푥) = {푔 · 푥 | 푔 ∈ 퐺} = {휙(푔) · 푥 | 푔 ∈ 퐺}. Since 휙 is surjective, we have {휙(푔) | 푔 ∈ 퐺} = 퐻. Hence, the orbit of 푥 under 퐺 is the same as that under

퐻: 풪퐺(푥) = 풪퐻 (푥) = {푕 · 푥 | 푕 ∈ 퐺}. Thus 퐺∖푋 = 퐻∖푋. Lastly, for 푥 ∈ {푃, 푄, 푒} (i.e. 푥 an element of 푇 ), we have the stabilizer in 퐺 of 푥 is,

퐺푥 = {푔 ∈ 퐺 | 푔 · 푥 = 푥}

= {푔 ∈ 퐺 | 휙(푔) · 푥 = 푥}

−1 = 휙 (퐻푥)

So by applying Theorem 2.2.3, we obtain the amalgam for 퐺.

We now state a theorem showing the connection between HNN extensions and groups acting on trees.

′ −1 Theorem 2.2.9. Suppose 퐺 = 퐺*퐴 = ⟨퐺, 푡 | 푡 푎푡 = 휃(푎), 푎 ∈ 퐴⟩, the HNN extension of 퐺 derived from the associated subgroups 퐴 and 휃(퐴) (Deﬁnition 2.1.11). Then there exists a unique (up to isomorphism) tree 푋 on which 퐺′ acts without inversion such that the quotient graph 퐺′∖푋 is a loop - a graph with one vertex and two mutually inverse edges.

Moreover, there is a segment 푇 in 푋 with vertex and edge stabilizers 퐺, 푡퐺푡−1, and 퐴, respectively. 56 Conversely, suppose 퐻 is a group acting without inversion on a tree 푌 such that the quotient 푓

graph 퐻∖푌 is a loop. Let 푍 = 푅 푆 be an arbitrary segment in 푌 . Let 푥 ∈ 퐻 be an

−1 arbitrary element such that 푆 = 푥푅 and let 휙 : 퐻푓 → 푥 퐻푓 푥 be an isomorphism induced by the

−1 conjugation by 푥. Then 푥 퐻푓 푥 ≤ 퐻푅 and the homomorphism

−1 (퐻푅)*퐻푓 = ⟨퐻푅, 푏 | 푏 푎푏 = 휙(푎), 푎 ∈ 퐻푓 ⟩ → 퐻

that sends the stable letter 푏 to 푥 and is the identity on 퐻푅, is an isomorphism.

For a proof of Theorem 2.2.9, see [Bog08, Ch.2, 15] and [Ser80, I.5] (although the proof there is more general, having to do with graph of groups - see below Section 2.3). Below we sketch the construction of the tree 푋 from the ﬁrst part of the theorem and comment on the second part.

Proof. Similar to Theorem 2.2.3, we deﬁne the graph 푋 by means of left cosets:

푉 푋 = 퐺′/퐺

ᨁ 퐸푋 = (퐺′/퐴) (퐺′/퐴)

where 훼(푔퐴) = 푔퐺 and 휔(푔퐴) = 푔푡퐺. It can be shown that 푋 is indeed a tree and clearly 퐺′ acts on 푋 by left multiplication and without inversion. The orbit of the vertex 퐺 under this action is 풪(퐺) = {푔′ · 퐺 | 푔; ∈ 퐺′} = 퐺′/퐺. Similarly, 풪(퐴) = 퐺′/퐴. Hence, we have that the quotient graph 퐺′∖푋 is a loop. Let 푇 be the segment with vertices equal to 퐺 and 푡퐺, with positively oriented edge 퐴. It is obvious that the stabilizers under the action of 퐺′ are 퐺, 푡퐺푡−1, and 퐴 respectively.

For the second part of the theorem, note that since 퐻푓 ≤ 퐻푅 and 푥 ∈ 퐻 was chosen so that 57 −1 푆 = 푥푅, we have for any 푥 푕푥 with 푕 ∈ 퐻푓 that:

(푥−1푕푥) · 푅 = (푥−1푕) · (푥푅)

= (푥−1푕) · 푆

= 푥−1 · (푕푆)

= 푥−1 · 푆 = 푅

−1 Thus, 푥 퐻푓 푥 ≤ 퐻푅. The rest of the theorem is proved in almost exactly the same way as Theorem 2.2.3 in [Bog08, 12.3]. Since we did not mention it before, let me describe roughly how one would show the

injectivity of the homomorphism 휓 :(퐻푅)*퐻푓 → 퐻.

The key is to use the existence of a tree for (퐻푅)*퐻푓 from the ﬁrst part of the theorem. Call

this tree 푋 and deﬁne a graph homomorphism 휓̃︀ : 푋 → 푌 by 푔퐻푧 ↦→ 휓(푔)푧 where 푧 ∈ {푅, 푆, 푓}

and 푔 ∈ (퐻푅)*퐻푓 . Showing that 휓 is surjective will imply that 휓̃︀ is surjective as well. Then one must show that 휓̃︀ is locally injective and use the following lemma:

Lemma 2.2.10. Let 푝 : 푋 → 푌 be a locally injective graph morphism from a connected graph 푋 to a tree 푌 . Then 푝 is in fact injective and 푋 is a tree.

One then uses that 휓̃︀ is an isomorphism to deduce the injectivity of 휓.

Example 2.2.11. In Example 1.2.9 we saw that the integers act on the tree 풞∞ with quotient graph a loop. Hence, we should have by Theorem 2.2.9 that the integers are isomorphic to an HNN extension. And indeed this is the case, as we saw in Example 2.1.12.

Example 2.2.12. The Baumslag-Solitar groups 퐵푆(푝, 푞) are HNN extensions (Example 2.1.13), thus by Theorem 2.2.9, each such group has an associated Bass-Serre tree 푋푝+푞, which is (푝 + 푞)−regular [DK, Example 4.56]. 58 2.3 Graphs of Groups

A central tool of Bass-Serre Theory is the concept of a graph of groups. Given a connected graph 푋 one assigns to each vertex 푣 ∈ 푉 푋, and each edge 푒 ∈ 퐸푋, a group. Precisely, we have:

Deﬁnition 2.3.1. A graph of groups 픾 = (풢, 푋) consists of a connected graph 푋 and a collection

of groups 풢 = {풢훾}, with 훾 ∈ 퐸푋 ∪ 푉 푋. The groups 풢푣, 푣 ∈ 푉 푋 are called vertex groups;

the 풢푒, 푒 ∈ 퐸푋 are called edge groups, with 풢푒 = 풢푒; and for each edge 푒 ∈ 퐸푋 there is a

monomorphism (injective morphism) 휙푒 : 풢푒 → 풢훼(푒).

풢푒

풢훼(푒) 풢휔(푒)

′ ′ Note that we also have monomorphisms 휙푒 : 풢푒 → 풢휔(푒), which are deﬁned by 휙푒 = 휙푒.

Example 2.3.2. Example 2.2.6 and Example 2.2.5 give us examples of graphs of groups. We

describe a graph of groups for SL2(ℤ). Let 푇 be a segment:

{푒, 푒} 푇 푃 푄

Declare vertex groups as 풢푃 = 푍4 and 풢푄 = 푍6 and edge groups 풢푒 = 풢푒 = 푍2. Let 푥 denote the

generator for 풢푃 and 푦 the generator for 풢푄. Then the monomorphisms 휙푒 and 휙푒′ are given by

2 3 푥 ↦→ 푥 and 푦 ↦→ 푦 , respectively. Thus we have that 픾 = (풢, 푇 ) where 풢 = {풢푃 , 풢푄, 풢푒}, is a graph of groups:

푍2 픾 = (풢, 푇 ) 푍4 푍6

Recall that one can form a fundamental group of a graph by simply realizing the graph as a CW-complex. In a (somewhat) similar fashion, we can form a fundamental group of a graph of groups. This we deﬁne below. 59 First, given a graph of groups 픾 = (풢, 푋), we deﬁne a group 퐹 (픾) (sometimes called the path group) generated by the vertex groups 풢푣 and the set of edges 퐸푋, subject to some relations:

−1 −1 ⟩︀ 퐹 (픾) = ⟨풢푣, 푒 ∈ 퐸푋| 푒 = 푒 , 푒 휙푒(푥)푒 = 휙푒(푥) , 푥 ∈ 풢푒

More precisely, let Γ0 be the free product of all the vertex groups:

Γ0 = * 풢푣 푣∈푉 푋

Then let 퐹 (퐸푋) be the free group on the set of edges,

퐹 (퐸푋) = 퐹 ({푒| 푒 ∈ 퐸푋}).

Now set Γ to be Γ0 * 퐹 (퐸푋). We factor Γ by the normal closure of the set

−1 −1 {푒푒, 푒 휙푒(푥)푒(휙푒(푥)) | 푒 ∈ 퐸푋, 푥 ∈ 풢푒}.

Thus,

 ⟨︀⟨︀ −1 −1 ⟩︀⟩︀ 퐹 (픾) = Γ 푒푒, 푒 휙푒(푥)푒(휙푒(푥)) | 푒 ∈ 퐸푋, 푥 ∈ 풢푒

We use the group 퐹 (픾) to form our fundamental group of 픾.

Deﬁnition 2.3.3. Let 픾 = (풢, 푋) be a graph of groups and ﬁx a vertex 푥0 in 푋. The fundamental group 휋1(픾, 푥0) of the graph of groups 픾 with respect to the vertex 푥0 is the subgroup of 퐹 (픾) generated by elements of the form

푔0푒1푔1푒2푔2 . . . 푒푛푔푛

where 푔0 ∈ 풢푥0 , 푔푖 ∈ 풢휔(푒푖 ), 푖 = 1, 2, . . . , 푛 and 푒1 . . . 푒푛 is a closed path in 푋 with initial vertex

푥0.

Example 2.3.4. With a somewhat trivial example, we can see how the fundamental group of a 60 graph of groups reverts to the fundamental group of the graph.

Suppose 픾 = (풢, 푋) is a graph of groups where all vertex groups are the trivial group, i.e.

−1 풢푣 = {1} for all 푣 ∈ 푉 푋. Then since 휙푒(풢푒) must be trivial, we have that 푒 휙푒(푥)푒 = 1. Hence,

 ⟨︀⟨︀ −1 ⟩︀⟩︀ 퐹 (픾) = 퐹 (퐸푋) 푒 = 푒 | 푒 ∈ 퐸푋

Fix a vertex 푥0 ∈ 푉 푋. The elements that form 휋1(픾, 푥0) are just closed paths in 푋, since there are no 푔’s to worry about. Because 푒 = 푒−1 in our path group, we have ensured that these paths ∼ are always reduced. Thus, we have that 휋1(픾, 푥0) = 휋1(푋, 푥0).

There is another way of defining the fundamental group of a graph of groups. Instead of using a fixed vertex in the graph, one fixes a maximal subtree.

Deﬁnition 2.3.5. Let 픾 = (풢, 푋) be a graph of groups and let 푇 be a maximal subtree in 푋. Then

the fundamental group 휋1(픾, 푇 ) of the graph of groups 픾 with respect to 푇 is the quotient group of 퐹 (픾) by the normal closure of the set {푒| 푒 ∈ 퐸푇 }. That is,

 휋1(픾, 푇 ) = 퐹 (픾) ⟨⟨푒| 푒 ∈ 퐸푇 ⟩⟩

We of course have a canonical homomorphism 푝 : 퐹 (픾) → 휋1(픾, 푇 )). It turns out, if we restrict 푝 to the subgroup 휋1(픾, 푥0) (assume we have ﬁxed some vertex 푥0), then 푝 is an isomorphism. And thus we have,

∼ Theorem 2.3.6 ([Bog08, 16.7]). 휋1(픾, 푥0) = 휋1(픾, 푇 ) for any choice of vertex 푥0 and any choice of maximal subtree 푇 .

Remark 2.3.7. So just as in the topology setting of fundamental groups, we have that the fundamental group of a graph of groups is independent of the choice of base point or maximal subtree.

Hence, when the context is clear we shall write 휋1(픾) to denote the fundamental group of the graph of groups 픾 61

Remark 2.3.8. Note that the groups 풢푣, 푣 ∈ 푉 푋 for a graph of groups 픾 = (풢, 푋), can be canonically embedded in the fundamental group 휋1(픾, 푇 ). For many details concerning this, see [Ser80, I.5.2] or [Bog08, Ch. 11, 14, 16].

푃 푒 푄 Example 2.3.9. Consider the graph 푋 = and suppose we have a graph of groups ∼ ∼ 픾 = ({풢푃 , 풢푒, 풢푄}, 푋). Then 휋1(픾, 푃 ) = 휋1(픾, 푋) = 풢푃 *풢푒 풢푄, the amalgamated free product

of 풢푃 with 풢푄 over the subgroups 휙푒(풢푒) and 휙푒(풢푒). Explicitly,

∼  휋1(픾) = (풢푃 * 풢푄) ⟨⟨휙푒(푡) = 휙푒(푡), 푡 ∈ 풢푒⟩⟩

∼ As an explicit example, let 픾 be as in Example 2.3.2. Then we have 휋1(픾) = 푍4 *푍2 푍6. Recall

from Example 2.2.6 that 푋 is (isomorphic to) the fundamental domain for SL2(ℤ) acting on a tree.

The previous example illustrates how graphs of groups and their fundamental groups give us a generalization of amalgamated free products. They also generalize the similar notion of HNN extension. 푒

푋

푃

Example 2.3.10. Consider the graph 푋 consisting of a single vertex 푃 and one edge pair {푒, 푒}. ∼ Let 픾 = ({풢푃 , 풢푒}, 푋) be a graph of groups. Then 휋1(픾, 푃 ) = 풢푃 *휙푒(풢푒), the HNN extension

′ of 풢푃 derived from the associated subgroups 휙푒(풢푒) and 휙푒(풢푒). The associated isomorphism

′ ′ −1 휃 : 휙푒(풢푒) → 휙푒(풢푒) is given by 휙 ∘ 휙 .

푝1 푝2 푝3 푥 푌 푇 푧 푦

푝4 푝5 62 Example 2.3.11. Consider the graph 푌 above with maximal subtree 푇 . Deﬁne a graph of groups

픾 = (풢, 푌 ) by 풢푝푖 = 푍 = ⟨푎푖⟩, 푖 ∈ {1, 2, 3, 4, 5, 6} and 풢푒 = {1} for all 푒 ∈ 퐸푌 . Since all the edge groups are trivial, we have each monomorphism 풢푒 → 풢훼(푒) is trivial.

The only edges not lying in 푇 are 푥, 푦, and 푧, thus the fundamental group 휋1(픾, 푌, 푇 ) is gen-

erated by the six generators 푎1, . . . , 푎6 of the six vertex groups and the edges 푥, 푦, and 푧. Since all

the edge groups are trivial, the only relations are 푒 = 푒−1 with 푒 ∈ 퐸푌 ∖ 퐸푇 . Hence, we have

∼ 6 휋1(픾) = (*푖=1ℤ) * 퐹,

where 퐹 is the free group with basis {푥, 푦, 푧}.

Example 2.3.12. More generally than the last example, suppose 푌 is a non-empty, connected graph and 픾 = (풢, 푌 ) a graph of groups with 풢푒 = {1} for all 푒 ∈ 퐸푌 . Let 퐴 be a chosen orientation for 푌 . Then:

휋1(픾) = (*푣∈푉 푌 풢푣) * 퐹, where 퐹 is the free group with basis 퐴 ∖ (푇 ∩ 퐴).

Example 2.3.13 ([Ser80, Remark, I.5.1, pg. 43]). We can get the same type of free group as the previous example via a quotient of the fundamental group of a graph of groups. Suppose we have a graph 푌 and a graph of groups 픾 = (풢, 푌 ). Choose a maximal subtree 푇 ⊂ 푌 and let 푅 be the smallest normal subgroup of 휋1(픾, 푌, 푇 ) that contains all the (images of the) vertex groups

풢푣, 푣 ∈ 푉 푌 . Then we must have that the quotient 휋1(픾)/푅 is given by

⟨︀푒, 푒 ∈ 퐸푌 | 푒 = 푒−1, 푒 = 1 if 푒 ∈ 퐸푇 ⟩︀ .

This is a free group with basis 퐴 ∖ (푇 ∩ 퐴), where 퐴 is an orientation for 푌 ; in other words,

휋1(픾)/푅 = 휋1(푌 ), the fundamental group of the graph 푌 .

The following definition will be very important in Section 3.5. 63 Definition 2.3.14. Let 픾 = (풢, 푋) be a graph of groups such that all the groups defined by 풢 are finite. Then we say 픾 is a finite graph of groups. We define the volume of a finite graph of groups 픾 to be ∑︁ 1 Vol(픾) = |풢푣| 푣∈푉 푋

2.4 Universal Covering Tree and Quotient Graph of Groups

In this section we examine what happens when a group acts on a tree. The results below are fundamental to Bass-Serre theory, and explain the amazing interplay between graphs of groups and trees. Our references throughout are [Bog08, Sec. 18] and [Ser80, I.5].

Universal Covering Tree

Firstly, suppose we have a graph of groups 픾 = (풢, 푌 ). Let 푇 be a maximal subtree of 푌 and let 퐺 = 휋1(픾, 푇 ), the fundamental group of this graph of groups with respect to 푇 . Then it turns out that there exists a tree 푋, called the universal covering tree of 픾, usually denoted 푋 = 픾̃︀ = (̃픾, 푇 ), such that 퐺 acts on 푋 without inversion, giving a quotient graph that is isomorphic to the graph 푌 . The theorem below details this in full.

Theorem 2.4.1 ([Bog08, 18.2]). Let 퐺 = 휋1(픾, 푇 ), where 픾 = (풢, 푌 ) is a graph of groups, 푇 a maximal subtree in 푌 . Then there exists a tree 푋 such that 퐺 acts without inversion of edges on 푋, and such that the quotient graph 퐺∖푋 is isomorphic to 푌 . Furthermore, the vertex and edge stabilizers of this action are conjugate to the canonical images in 퐺 of the vertex groups

풢푣, 푣 ∈ 푉 푌 and 휙푒(풢푒), 푒 ∈ 퐸푌 , respectively. Additionally, for the projection 푝 : 푋 → 푌 ∼= 퐺∖푋, there exists a lift of the pair (푇, 푌 ),

denoted (푇̃︀ , 푌̃︀), such that

1. the stabilizer 퐺푣̃, 푣˜ ∈ 푉 푇̃︀ is equal to 풢푣;

2. if 푒˜ ∈ 퐸푌̃︀ with 훼(˜푒) ∈ 퐸푇̃︀, then the stabilizer 퐺푒̃ = 풢푒;

3. if 푣 = 휔(˜푒) ∈/ 푉 푇̃︀, then the element 푒−1 ∈ 퐺 carries 푣 into 푉 푇̃︀ under the action of 퐺. I.e., if 푣 = 휔(˜푒) ∈/ 푉 푇̃︀, then 푒−1푣 ∈ 푉 푇̃︀. 64 Details of the proof of this theorem can be found in [Bog08], [Ser80, I.5.3], and [DD89, I.7].

Bass in [Bas93] gives a proof for 픾 defined via a fixed vertex. But so as to not leave the reader completely in the dark, below we show how one defines the universal covering tree 푋.

First, choose some orientation for 푌 (this is mostly just for ease). Identify the groups 풢푣 and

풢푒 with their canonical images in 퐺, noting that 풢푒’s image is actually that of 휙푒(풢푒). Now we can deﬁne vertices and edges by taking left cosets in 퐺:

ᨁ  ᨁ  푉 푋 = 퐺 풢푣, 퐸푋+ = 퐺 풢푒

푣∈푉 푌 푒∈퐸푌+

and where 훼(푔풢푒) = 푔풢훼(푒) and 휔(푔풢푒) = 푔푒풢휔(푒). We have 퐺 act on 푋 via left multiplication. One can easily verify that this is indeed a proper graph action. One thing to check is that, if 푕 ∈ 퐺, then 푕 · 훼(푔풢푒) = 훼(푕 · 푔풢푒). But by our deﬁnition, we see that

푕 · 훼(푔풢푒) = 푕 · 푔풢훼(푒)

= (푕푔)풢훼(푒)

= 훼(푕푔풢푒) = 훼(푕 · 푔풢푒)

Similarly, 푕 · 휔(푔풢푒) = 휔(푕 · 푔풢푒), and since 풢푒 = 풢푒, we have 푕 · 푔풢푒 = 푕 · 푔풢푒. One can see by our deﬁnition (the details are trivial) that we indeed have an isomorphism ∼ 퐺∖푋 = 푌. For 푔풢푦 ∈ 푋, with 푦 ∈ 푌 , the isomorphism is simply 풪(푔풢푦) ↦→ 푦. How exactly is

the lift (푇̃︀ , 푌̃︀) deﬁned? For 푇̃︀, we simply have all the identity cosets making up the vertices and edges: ᨁ ᨁ 푉 푇̃︀ = {풢푣}, 퐸푇̃︀+ = {풢푒}

푣∈푉 푇 푒∈퐸푇+

And then 푌̃︀ is deﬁned as follows:

ᨁ ᨁ 푉 푌̃︀ = 푉 푇̃︀ ∪ {푒풢휔(푒)}, 퐸푌̃︀ = 퐸푇̃︀ ∪ {풢푒} 푒∈퐸푌 −퐸푇 푒∈퐸푌 −퐸푇 65

Clearly 푇̃︀ ⊂ 푌̃︀. If 푒˜ ∈ 퐸푌̃︀+ − 퐸푇̃︀+, then 푒˜ is of the form 풢푒, 푒 ∈ 퐸푌+ − 퐸푇+. Thus, 훼(˜푒) = 풢훼(푒) and 휔(˜푒) = 푒풢휔(푒), meaning this terminal vertex lies in 푇̃︀. For the inverse edge, one will have the initial edge lie in 푇̃︀, and thus (푇̃︀ , 푌̃︀) satisﬁes the second condition of being a lift of the pair (푇, 푌 ). The third condition can also be easily checked.

Quotient Graph of Groups

Above we had a graph of groups yield a universal covering tree on which the fundamental group of the graph of groups acted, yielding the original graph. Now suppose the converse: let 퐺 be a group acting without inversion of edges on a tree 푋. Let 푌 = 퐺∖푋 be the quotient graph with projection 푃 : 푋 → 푌 . Furthermore, let 푇 be a maximal subtree in 푌 with (푇̃︀ , 푌̃︀) a lift of the pair (푇, 푌 ). The result below tells us that, there exists a graph of groups 픾 = (풢, 푌 ), called a quotient graph of groups, such that our original group 퐺 is isomorphic to 휋1(픾, 푇 ). One deﬁnes the graph of groups 픾 = (풢, 푌 ) as follows. Recall that (푇̃︀ , 푌̃︀) a lift of (푇, 푌 ) means 푝 : 푇̃︀ → 푇 is an isomorphism and 푝 : 퐸푌̃︀ − 퐸푇̃︀ → 퐸푌 − 퐸푇 is a bijection. Because of this, we always have lifts for any 푦 ∈ 푌 .

Let 퐺푦̃ denote the stabilizer in 푋 of the lift of 푦 ∈ 푌 . For our graph of groups, we set 풢푦 = 퐺푦̃. That is,

풢푣 = 퐺푣̃ for 푣 ∈ 푉 푌

풢푒 = 퐺푒̃ for 푒 ∈ 퐸푌

Since in the fundamental group of a graph of groups with respect to a maximal subtree 푇 all the edges in 푇 equal 1, we must properly deﬁne the edges in our subtree 푇 , as well as those on the boundary of 푌 , i.e. in 퐸푌 − 퐸푇. This will ensure we get the desired isomorphism. So suppose

푒 ∈ 퐸푌 − 퐸푇 with 휔(˜푒) ∈/ 푉 푇̃︀. We may choose an element 푔푒 ∈ 퐺 so that 휔(˜푒) = 푔푒휔̃︂(푒). Why?

Because 휔(˜푒) and 휔̃︂(푒) have the same projection in 푌 , hence lie in the same 퐺 orbit. So such a 푔푒 must exist.

−1 If 푒 ∈ 퐸푇 , then such a 푔푒 will just be 1. Set 푔푒 = 푔푒 . We can now deﬁne the monomorphisms 66 ′ ′ necessary for 픾 = (풢, 푌 ). For each 푒 ∈ 퐸푌, deﬁne 휙푒 : 풢푒 → 풢휔(푒) (recall 휙푒 = 휙푒) by the following formula: ⎧ ⎪ ⎪푕 if 휔(˜푒) ∈ 푉 푇̃︀ ′ ⎨ 휙푒(푕) = ⎪ −1 ⎩⎪푔푒 푕푔푒 if 휔(˜푒) ∈ 푉 푌̃︀ − 푉 푇̃︀

−1 One thing to check here is that we have 푔푒 풢푒푔푒 ⊂ 풢휔(푒). But this is very straight forward by our

choice of 푔푒. Suppose 푒 ∈ 퐸푌 with 휔(˜푒) ∈/ 푉 푇̃︀. Let 푕 ∈ 풢푒. Then 푕 stabilizes 푒˜, and hence 휔(˜푒). Thus,

−1 −1 푔푒 푕푔푒 · 휔̃︂(푒) = 푔푒 푕휔(˜푒)

−1 = 푔푒 휔(˜푒)

= 휔̃︂(푒)

−1 and so 푔푒 풢푒푔푒 stabilizes 휔̃︂(푒), i.e. lies in 풢휔(푒).

We have now fully deﬁned our quotient graph of groups 픾 = (풢, 푌 ). Note, 픾 depends on the

lift (푇̃︀ , 푌̃︀) and the choice of elements {푔푒}푒∈퐸푌 . As in [BL01], we will often denote 픾 by 퐺∖∖푋.

Now we deﬁne a homomorphism 휑 : 휋1(픾) → 퐺 by

휑(푔) = 푔, for 푔 ∈ 풢푣, 푣 ∈ 퐸푌

−1 휑(푒) = 푔훼(푒)푔휔(푒), for 푒 ∈ 퐸푌

We now state or second fundamental result of Bass-Serre theory.

Theorem 2.4.2 ([Ser80, I.5.4, Theorem 13]). Let 퐺 be a group acting without inversion on a tree 푋. Let 퐺∖∖푋 be the quotient graph of groups and 휑 : 퐺∖∖푋 → 퐺, as deﬁned above. Then 휑 is a group isomorphism.

Using the map 휑, we can deﬁne a map 휓 : 픾̃︀ → 푋, where 픾̃︀ denotes the universal covering 67 tree of our quotient graph of groups. The map 휓 is deﬁned by:

휓(푔풢푣) = 휑(푔)˜푣 and 휓(푔풢푒) = 휑(푔)˜푒 where 푣 ∈ 푉 푌 and 푒 ∈ 퐸푌 . Then we have that:

Corollary 2.4.3 ([Ser80, I.5.4, Theorem 13]). The map 휓 : 픾̃︀ → 푋 deﬁned above is a graph isomorphism.

Remark 2.4.4. To summarize a little, the above theory tells us that, if we have a connected graph 푋, and a group 퐺 acting without inversion (which, recall we can assume without loss of generality for any group and graph), then 퐺 and 푋 can be identiﬁed, respectively, with the fundamental group

휋1(픾) of the quotient graph of groups 픾 = (퐺∖푋, 풢) = 퐺∖∖푋, and the universal covering tree 푋̃︀ = 픾̃︀. Furthermore, via this identiﬁcation, if 푒 ∈ 퐸푋 with 훼(푒) = 푥, and 푝(푥) = 푦, 푝(푒) = 푓 with 푝 : 푋 → 퐺∖푋 the projection map, then the monomorphism 휙푓 : 풢푓 → 풢푦 is identiﬁed with

the inclusion map 퐺푒 → 퐺푥 between group stabilizers in 퐺. Moreover,

⃒ ⃒ ⃒(︀ )︀−1 ⃒ [풢푦 : 휙푓 (풢푓 )] = [퐺푥 : 퐺푒] = ⃒ 푝|St(푥) (푓)⃒ = |퐺푥 · 푒|

2.5 Nagao’s Theorem

In this section, we apply the Bass-Serre theory developed in the previous sections to show the following:

Theorem 2.5.1. The group GL2(푘[푡]) can be written as the following amalgamated free product:

GL2(푘[푡]) = GL2(푘) *퐵(푘) 퐵(푘[푡])

⎧⎛ ⎞⃒ ⎫ ⃒ ⎨⎪ 푎 푏 ⃒ ⎬⎪ ⎜ ⎟⃒ * where 퐵(푅) = ⎝ ⎠⃒ 푎, 푑 ∈ 푅 , 푏 ∈ 푅 for a general ring 푅, and 푘 a commutative ﬁeld. ⎪ 0 푑 ⃒ ⎪ ⎩ ⃒ ⎭ Nagao[Nag59] proves this directly, but Serre[Ser80] gives a proof using the Bruhat-Tits tree 68 and Bass-Serre theory. The advantage to the latter method is that it can be used more generally for other rings, not just 푘[푡] (cf. [Ser80, II.2]).

Let 퐾 = 푘(푡) and Γ = GL2(푘[푡]). We place on 퐾 the valuation 휈 = 휈∞ from Example 1.3.5. Hence (︂푓 )︂ 휈 = deg(푔) − deg(푓) 푔 {︂ ⃒ }︂ 푓 ⃒ and the valuation ring 풪 = ∈ 퐾⃒ deg(푔) ≥ deg(푓) . 푔 ⃒ 2 푛 Let 푉 = 퐾 with canonical basis {푒1, 푒2}. For 푛 ≥ 0, set 퐿푛 = 풪(푡 푒1)⊕풪푒2 and [퐿푛] = Λ푛, the corresponding lattice class. Let 푋 be the Bruhat-Tits tree associated with 푉 Section 1.3 (cf. Section 1.3).

The Λ푛’s form a reduced path in 푋:

Λ0 Λ1 Λ2 Λ3

We set Γ0 = GL2(푘) and for 푛 ≥ 1,

⎧⎛ ⎞⃒ ⎫ ⃒ ⎨⎪ 푎 푏 ⃒ ⎬⎪ ⎜ ⎟⃒ * Γ푛 = ⎝ ⎠⃒ 푎, 푑 ∈ 푘 , 푏 ∈ 푘[푡], deg(푏) ≤ 푛 ⎪ 0 푑 ⃒ ⎪ ⎩ ⃒ ⎭

Notice that Γ0 ≤ Γ1 ≤ Γ2 ≤ ... . Now we have the following from the action of Γ on the tree 푋:

Proposition 2.5.2 ([Ser80, II.1.6, Prop. 3]).

1. The Λ푛 are pairwise inequivalent mod Γ (i.e. in the quotient graph).

2. Γ푛 is the stabilizer of Λ푛 in Γ.

3. Γ0 acts transitively on the set of edges in 푆푡(Λ0).

4. For 푛 ≥ 1, Γ푛 ﬁxes the edge 푒Λ푛Λ푛+1 , and acts transitively on the set of edges in 푆푡(Λ푛) that

are distinct from 푒Λ푛Λ푛+1 . 69 Recall that a group 퐺 acts transitively on a set 푆 if, for any 푠, 푠′ ∈ 푆, there exists 푔 ∈ 퐺 such that 푔푠′ = 푠. A full proof can be found in [Ser80] (see also our own discussion of vertex and edge stabilizers in Chapter 4). The following lemma (along with Proposition 2.5.2) will allow us to prove Theorem 2.5.1:

Lemma 2.5.3 ([Ser80, II.1.6, Corollary]).

Λ0 Λ1 Λ2 Λ3 The path 푇 = is a fundamental domain

for 퐺∖푋.

Proof of Theorem 2.5.1. We have Γ acting on our tree 푋 and the quotient graph is isomorphic to the above path 푇 . Hence, we can form a quotient graph of groups 픾 = (푇, 풢). The groups here

will just be the stabilizers of the vertices and edges in 푋, which are all the Γ푖’s.

Γ0 Γ1 Γ2 Γ3

Γ0 Γ1 Γ2

Thus, we will obtain for 휋1(픾) a free product of all the vertex groups with each other, amalgamated over the edge groups. The Γ푛’s form an increasing chain of subgroups, Γ1 ≤ Γ2 ≤

Γ3 ≤ ... . Hence, we will obtain their union when we take their amalgamated free product (cf.Example 2.1.10): ⋃︁ * Γ푛 = Γ푛 = 퐵(푘[푡]). 푛≥1 푛≥1

Lastly, we must amalgam along the very ﬁrst segment. Since Γ0 ∩ Γ1 = GL2(푘) ∩ Γ1 = 퐵(푘), we have in total that

Γ = GL2(푘) *퐵(푘) 퐵(푘[푡])

Let 푆퐵(푅) = SL2(푅) ∩ 퐵(푅), for 퐵(푅) deﬁned as in Theorem 2.5.1. Then we have a similar

result as above, but for SL2(푘[푡]). 70

Corollary 2.5.4. SL2(푘[푡]) = SL2 *푆퐵(푘)푆퐵(푘[푡])

This is easily proved using the following lemma and Theorem 2.5.1:

Lemma 2.5.5. Suppose 퐺 = 퐺1 *퐴 퐺2 is an amalgamated free product of groups 퐺1 and 퐺2 over a common subgroup 퐴. Let 퐻 ≤ 퐺 and assume that AH=G. Set 퐵 = 퐴 ∩ 퐻 and 퐻푖 = 퐺푖 ∩ 퐻 for 푖 = 1, 2. Then 퐻 = 퐻1 *퐵 퐻2.

Lastly,

Corollary 2.5.6 ([Nag59, Nagao]). The groups GL2(푘[푡]) and SL2(푘[푡]) are not ﬁnitely generated.

This is quite interesting since, for one, it holds for any ﬁeld 푘. Also, for 푘 ﬁnite, this is no longer true for 퐺퐿푛(푘[푡]) with 푛 ≥ 3, as these groups have Khazdan’s property (T) [BdlHV08].

2.6 Edge-Indexed Graphs

In Section 2.4, we saw how when a group 퐺 acts on a tree 푋 without inversions, 퐺 and 푋 can be identiﬁed with the fundamental group and the universal covering tree, respectively of the quotient graph of groups 퐺∖∖푋. Many results in the realm of tree lattices involve obtaining a graph of groups, for which we can then apply the above theory to better understand the group 퐺. The method of edge-indexing a graph, deﬁned below, helps us obtain a graph of groups from an arbitrary connected graph.

Deﬁnition 2.6.1. Let 픸 = (풜, 퐴) be a graph of groups. For each 푒 ∈ 퐸퐴, 훼(푒) = 푎, deﬁne

푖풜(푒) = 푖(푒) := [풜푎 : 휙푒(풜푒)] called the index of the edge 푒. Deﬁne 퐼(픸) := (퐴, 푖), the edge-indexed graph associated to 픸. If given such a (퐴, 푖), a graph of groups 픸 that satisﬁes

푖(푒) = [풜푎 : 휙푒(풜푒)] 71 for every edge 푒 ∈ 퐸퐴 with 훼(푒) = 푎, is called a grouping of (퐴, 푖). We say 픸 is a ﬁnite grouping for (퐴, 푖) if 픸 is a graph of ﬁnite groups.

Remark 2.6.2. When using a diagram to describe a graph, we will sometimes explicitly give an indexing by placing the values 푖(푒) on edges. For an oriented edge 푒, we will place 푖(푒) closest to 훼(푒) and 푖(푒) closest to 휔(푒). See the ﬁgure below.

푖(푒) 푖(푒)

Figure 2.1: Schema for Edge-Indexed Graphs

In order to get the full force of the Bass-Serre theory, we sometimes require a graph of groups,

such as 픸, to be effective or faithful:

Deﬁnition 2.6.3. Let 픸 = (풜, 퐴) be a graph of groups, 푎0 ∈ 퐴 a ﬁxed vertex, and 푋 = (̃픸, 푎0)

the universal covering tree. If Γ = 휋1(픸, 푎0) acts faithfully on 푋, i.e. the action deﬁning homomorphism Γ → 푋 is injective, then we say say the graph of groups 픾 is faithful or effective.

Fix a basepoint 푎0 ∈ 퐴. Then the universal covering tree 푋 = (̃픸, 푎0), with projection map

푝 : 푋 → 퐴, depends up to isomorphism only on (퐴, 푖, 푎0), the edge-indexed graph associated to

픸 with ﬁxed basepoint, and not the grouping itself (cf. Section 1.2 and [Bas93, 1.18]). Because of

this fact, we also denote 푋 = (퐴,̃ 푖, 푎0).

Now we can utilize the fundamental results of the Bass-Serre theory Section 2.4 above. If 픸 is

a grouping for an edge-indexed graph (퐴, 푖), then the group Γ = 휋1(픸, 푎0), 푎0 ∈ 퐴 acts without

inversion of edges on the universal covering tree 푋 = (̃픸, 푎0) = (퐴,̃ 푖, 푎0), with quotient graph Γ∖푋 = 퐴. Nicely enough, we can assume a graph of groups is effective without too much loss

of generality. This is because any graph of groups 픸 yields an effective quotient, 픸′ = (퐴, 풜′),

′ ′ deﬁned by taking quotients of groups in 풜. That is, groups 풜푣 and 풜푒 as quotients of 풜푣 and 풜푒, respectively, where 푣 ∈ 푉 퐴, 푒 ∈ 퐸퐴. One also must insist that the edge monomorphisms deﬁned

′ for 픸 , denoted 휙̂︁푒 yields the following commutative diagram: 72 휙푒 풜푒 풜푣

′ 휙̂︁푒 ′ 풜푒 풜푣

One property to note here is that the faithful quotient has no effect on edge-indexing: 퐼(픸) = 퐼(픸′). For all the details of an effective quotient, see [Bas93, 1.24, 1.25]

Now we return to our case above, of 픸 a grouping for (퐴, 푖). By possibly taking the faithful

′ quotient of 픸 , we can identify Γ = 휋1(픸, 푎0) with its image in 퐺 = Aut(푋). Then the vertex and edge stabilizers in Γ are isomorphic to the vertex and edge groups of 픸′.

2.7 Properties of Edge-Indexed Graphs

Suppose we have an edge-indexed graph (퐴, 푖) (Definition 2.6.1). How do we know if there is a grouping for (퐴, 푖)? In particular, a finite grouping? The following theorem gives us equivalent statements that answer this question. The terms involved shall be defined after the statement of the theorem (which is coming directly from [BL01, 2.6(19)]).

Theorem 2.7.1. Let (퐴, 푖) be an edge-indexed graph, where 퐴 is connected and the indices 푖(푒)

are ﬁnite: 푖 : 퐸퐴 → ℤ with 푖(푒) > 0 for all 푒 ∈ 퐸퐴. Then the following are equivalent:

1. (퐴, 푖) has bounded denominators (thus is unimodular).

푁(훼(푒)) 2. There exists a function 푁 : 푉 퐴 → {0} deﬁned on edges of 퐴 by 푁(푒) = , ℤ ∖ 푖(푒) such that 푁(푒) = 푁(푒) for all 푒 ∈ 퐸퐴.

3. (퐴, 푖) admits a ﬁnite grouping.

The third equivalence was our question, and thus one and two give an answer. We now explain the terms bounded denominators and unimodular.

× Using our index function 푖, we deﬁne a function Δ: 퐸퐴 → ℚ>0 by

푖(푒) Δ(푒) = 푖(푒) 73 × −1 (Note: ℚ>0 means the multiplicative group of positive rational numbers.) Clearly, Δ(푒) = Δ(푒) .

If 푙 is a path in 퐴, 푙 = 푒1 . . . 푒푛, then we deﬁne Δ on paths by

푛 ∏︁ Δ(푙) := Δ(푒푖) 푖=1

Note that if 푙 is the trivial path, then Δ(푙) = 1. Let 푙 denote the reverse path of 푙. If 푙 = 푒1 . . . 푒푛,

−1 then 푙 = 푒푛 ... 푒1. We have Δ(푙) = Δ(푙) .

Deﬁnition 2.7.2. Let (퐴, 푖) be an edge-indexed graph of ﬁnite index (푖(푒) > 0 for all 푒 ∈ 퐸퐴).

× Let Δ: 퐸퐴 → ℚ>0 be deﬁned as above. Then if Δ(푙) = 1 for all closed paths 푙, we say (퐴, 푖) is unimodular.

Lemma 2.7.3. Let 푎, 푏 ∈ 푉 퐴 and 푙 a path from 푎 to 푏. If (퐴, 푖) is unimodular, then the rational number Δ(푙) depends only on the vertices 푎 and 푏 and not on the path 푙.

Proof. Let 푙1 = 푒1 . . . 푒푛 and 푙2 = 푓1 . . . 푓푚 be distinct paths in 퐴 from 푎 to 푏. Then the path

푙1푙2 = 푒1 . . . 푒푛푓푚 ... 푓1

−1 is closed in 퐴. Hence, by unimodularity, Δ(푙1푙2) = 1. Since Δ(푙2) = Δ(푙2) , we see that

Δ(푙1푙2) = 1

−1 Δ(푙1)Δ(푙2) = 1

Δ(푙1) = Δ(푙2)

In light of the above lemma, assuming that (퐴, 푖) is unimodular, we can define Δ on vertices by Δ(푏) := Δ(푙), where 푙 is any path from 푎 to 푏 in 퐴. The following definition is now well-defined Δ(푎) because of our lemma: 74 Definition 2.7.4. Let (퐴, 푖) be an edge-indexed graph of finite index and suppose (퐴, 푖) is unimodular. Fix a vertex 푎 ∈ 푉 퐴. We define the volume at base point 푎 by:

∑︁ 1 ∑︁ Δ(푎) Vol ((퐴, 푖)) = = 푎 Δ(푙) Δ(푏) 푙 a path from 푏∈푉 퐴 푎 to 푏

Note that if 0 < Δ(푎′) and Δ(푎) < ∞ then

(︂Δ(푎′))︂ Vol ′ ((퐴, 푖)) = Vol ((퐴, 푖)). 푎 Δ(푎) 푎

Thus, when the volume is ﬁnite we can write

Vol((퐴, 푖)) < ∞.

Definition 2.7.5. Let (퐴, 푖) be an edge-indexed graph with finite indexing. For 푎 ∈ 푉 퐴, define

{︂ ⃒ }︂ Δ(푙) ×⃒ 푅푎 = ∈ ℚ ⃒ 푒 ∈ 퐸퐴, 푙 is a path from 푎 to 훼(푒) 푖(푒) ⃒

If for some 푎 ∈ 푉 퐴 (and hence for all vertices), 푅푎 has bounded denominators, then we say (퐴, 푖) has bounded denominators.

Lemma 2.7.6. If (퐴, 푖) has bounded denominators, then (퐴, 푖) is unimodular.

Proof. We shall do a proof by the contrapositive. Suppose (퐴, 푖) is not unimodular. So there exists a closed path 푙, say at vertex 푎, with Δ(푙) ̸= 1. Without loss of generality, assume Δ(푙) > 1 (we can replace 푙 with 푙 if this is not the case). For any edge 푒 ∈ 푆푡(푎), 푙 is a path from 푎 to 훼(푒) = 푎. In particular, 푙푛 is such a path for any 푛 > 0. But then the collection of rational numbers

{︂ 푛 푛 ⃒ }︂ Δ(푙 ) Δ(푙) ⃒ = ⃒ 푒 ∈ 푆푡(푎), 푛 > 0 푖(푒) 푖(푒) ⃒

is unbounded, hence has unbounded denominators, and is obviously a subset of 푅푎. Therefore 75 (퐴, 푖) does not have bounded denominators.

So an edge-indexed graph (퐴, 푖) having bounded-denominators implies unimodularity, as The- orem 2.7.1 mentioned. Notice also that, if 퐴 is a ﬁnite graph, then we necessarily have bounded denominators. So in the presence of a ﬁnite graph, this is an if and only if. In light of this, we have a corollary to Theorem 2.7.1:

Corollary 2.7.7. If in (퐴, 푖) the graph 퐴 is ﬁnite, then (퐴, 푖) admits a ﬁnite grouping if and only if (퐴, 푖) is unimodular.

Now suppose (퐴, 푖) has a ﬁnite grouping, 퐼(픸) = (퐴, 푖). Then for 푒 ∈ 퐸퐴 with 훼(푒) = 푎,

|풜푎| 푖(푒) = [풜푎 : 휙(풜푒)] = |풜푒|

Then we see that for an edge 푒 with 훼(푒) = 푎, 휔(푒) = 푏

푖(푒) Δ(푒) = 푖(푒) |풜 | |풜 | = 푏 · 푒 |풜푒| |풜푎| |풜 | = 푏 |풜푎|

So for two vertices 푎, 푏 ∈ 푉 퐴, with 푙 = 푒1 . . . 푒푛 a path from 푎 to 푏,

푛 ∏︁ Δ(푙) = Δ(푒푖) 푖=1 |풜 | |풜 | |풜 | = 훼(푒2) · 훼(푒3) ... 푏 |풜푎| |풜훼(푒2)| |풜훼(푒푛−1)| |풜 | = 푏 |풜푎|

Hence, we can choose that Δ(푎) = |풜푎| for any vertex 푎 ∈ 푉 퐴. 76 We can relate our two separate deﬁnitions of volume via the equation:

∑︁ |풜푎| Vol푎((퐴, 푖)) = = |풜푎| · Vol(픸) |풜푏| 푏∈퐸퐴

The details of the proof for Theorem 2.7.1 are highlighted in [BL01, 3.6], and given proper detail in [BK90, Sections 1 & 2]. There you can ﬁnd more information about this mysterious function 푁 mentioned above, as well as a canonical inﬁnite cyclic grouping for (퐴, 푖).

Example 2.7.8 (From [BK90, 1.5]). Consider the following edge-indexed graph (퐴, 푖):

푚 (퐴, 푖) = 푎 푒 푛

where 푖(푒) = 푛 and 푖(푒) = 푚, where 푛 and 푚 are positive integers. Is (퐴, 푖) unimodular? The only closed paths in 퐴 are (푒) and (푒). We have

푚 푛 Δ(푒) = , Δ(푒) = 푛 푚

Thus (퐴, 푖) is unimodular if and only if 푛 = 푚. By the above Corollary 2.7.7, (퐴, 푖) admits an inﬁnite grouping if and only if 푛 = 푚.

Example 2.7.9. Continuing with (퐴, 푖) from Example 2.7.8 above, we give an example of a grouping.

Let 풜푎 = 풜푒 = ℤ. Deﬁne 휙푒 : ℤ → ℤ by 휙푒(푥) = 푛 · 푥. Then 휙푒(ℤ) = 푛ℤ and

푖(푒) = [ℤ : 푛ℤ] = 푛

We similarly deﬁne 휙푒 : ℤ → ℤ by 휙푒(푥) = 푚 · 푥, so that 휙푒(ℤ) = 푚ℤ and

푖(푒) = [ℤ : 푚ℤ] = 푚 77 The graph of groups 픸 = (퐴, 풜) now has 퐼(픸) = (퐴, 푖), but is not a finite grouping. The above procedure can be generalized to any graph (퐴, 푖) to give the canonical infinite cyclic grouping, as in [BK90, (2.2)]. It turns out that, if we take the effective quotient of (퐴, 푖) defined here, then we will obtain a finite grouping. However, this is rather complicated, so we will leave it alone.

Example 2.7.10. Again, let (퐴, 푖) be the edge-indexed graph deﬁned in Example 2.7.8. Here we give a ﬁnite grouping of (퐴, 푖), thus we set 푛 = 푚.

We deﬁne a graph of groups 픸 = (퐴, 풜) by

풜푎 = ℤ2푛

the cyclic group of order 2푛, and

풜푒 = 풜푒 = ℤ2

the cyclic group of order 2. We write ℤ2 = {0, 1} and ℤ2푛 = ⟨푥⟩. We deﬁne the monomorphisms

′ 휙푒, 휙푒 : ℤ2 → ℤ2푛 by ′ 푛 휙푒(1) = 휙푒(1) = 푥

Thus 픸 is a ﬁnite graph of groups. One can see that [ℤ2푛 : 휙푒(ℤ2)] = 푛, hence 퐼(픸) = (퐴, 푖), and so 픸 is a ﬁnite grouping for (퐴, 푖).

Example 2.7.11 (Also from [BK90, 1.5]). This examples shows how, when 퐴 is not a ﬁnite graph, unimodular does not force bounded denominators. Consider the graph 퐴 below.

푒1 푒2 푒3 푒푛 퐴 = 푥1 푥2 푥3 푥4 푥푛

We give two indexed graphs of 퐴. First is (퐴, 푖) where 푖(푒푛) = 1 and 푖(푒푛) = 2, for 푒푛 ∈ 퐸퐴, for

all 푛 ≥ 1. The second will be denoted (퐴, 푗), where 푗(푒푛) = 2 and 푗(푒푛) = 1 for 푒푛 ∈ 퐸퐴, for all 푛 ≥ 1. For (퐴, 푖), we do get bounded denominators. But in (퐴, 푗) we do not.

To see why, let 푎 = 푥1. Then from the deﬁnition of bounded denominators, we want to know 78 if the denominators in

{︂ ⃒ }︂ Δ(푙) ×⃒ 푅푎 = ∈ ℚ ⃒ 푒 ∈ 퐸퐴, 푙 is a path from 푎 to 훼(푒) 푖(푒) ⃒

are bounded. For any 푒푛 ∈ 퐸퐴, any path from 푎 to 훼(푒푛) will reduce to one of the form 푙 =

푒1 . . . 푒푛. So these are the only paths we need to bother computing with. 2 For (퐴, 푖), we have Δ(푒 ) = . Thus for a path 푙 = 푒 . . . 푒 from 푎 to 훼(푒 ), 푛 1 1 푛 푛

푛 ∏︁ 푛 Δ(푙) = Δ(푒푖) = 2 푖=1

Δ(푙) 2푛 Hence, the rational numbers in 푅푎 will all be of the form . Thus we have bounded 푖(푒푛) 1 denominators in (퐴, 푖). 1 For (퐴, 푗), Δ(푒 ) = . Hence for a path 푙 = 푒 . . . 푒 from 푎 to 훼(푒 ), 푛 2 1 푛 푛

푛 ∏︁ 1 Δ(푙) = Δ(푒 ) = 푖 2푛 푖=1

1 Since 푗(푒 ) = 2, the numbers in 푅 will be of the form . Clearly these denominators become 푛 푎 2푛+1 unbounded as 푛 → ∞, hence (퐴, 푗) does not have bounded denominators. One last thing to mention: notice that since 퐴 is a tree, there are no non-trivial, closed, reduced paths. Thus, any closed path 푙 in 퐴 will have Δ(푙) = 1, for any indexing of 퐴. So unimodularity always holds for 퐴, and so we see in the case of (퐴, 푗) above, we have unimodularity but not bounded denominators.

2.8 Automorphism Types

Here we detail the possible forms that a tree automorphism can have. Since we are always assuming that 퐺 acts on 푋 without inversion of edges, we only have two different types of automorphisms in 퐺: elliptic and hyperbolic. Recall that automorphisms preserve the lengths of geodesics. Because of this, we can define: 79 Definition 2.8.1. Let 푔 ∈ 퐺. The translation length of 푔 is defined as:

|푔| = inf 푑(푣, 푔(푣)) 푣∈푉 푋

Here, 푑 denotes the path-metric in 푋.

If |푔| = 0, then 푔 must ﬁx a point in 푋. If |푔| > 0, then 푔 ﬁxes no points. In a tree, we have that any 푔 ∈ 퐺 will always attain its infemum; thus we can always write:

|푔| = min 푑(푣, 푔(푣)) 푣∈푉 푋

The set of points where 푔 attains |푔| is a subtree of 푋, and is denoted 푇푔. That is, the minimal subtree of 푋 that contains all vertices satisfying 푑(푣, 푔(푣)) = |푔| :

푉 푇푔 = {푣 ∈ 푉 푋 | 푑(푣, 푔(푣)) = |푔|}

We have some special notations and names:

Deﬁnition 2.8.2. If |푔| = 0, then 푔 is called an elliptic automorphism or sometimes simply a ∘ rotation. We deﬁne 푇푔 := 푇푔 in this case. → For |푔| > 0, we say 푔 is a hyperbolic automorphism or a translation. We write 푇푔 := 푇푔 which in this case is called an axis of 푔 or 푔-axis.

Example 2.8.3. Recall that Aut(풞∞) = 퐷∞ (Example 1.1.9) and every automorphism is of the

푛 푛 form 푏 푎 or 푏 for 푛 ∈ ℤ. The automorphism 푎 is deﬁned by 푎(푒0) = 푒−1 and hence 푎(훼(푒0)) =

훼(푒−1) = 훼(푒0); so 푎 ﬁxes the vertex 0. Thus |푎| = 0 and 푎 is an elliptic automorphism.

푛 푛 The automorphisms 푏 , 푛 ∈ ℤ act as translations on 풞∞, sending 푒0 to 푒푛. Hence |푏 | = |푛|

and these are all hyperbolic automorphisms. The axis 푇푏푛 is clearly the entire tree 풞∞.

The Theorem below (due to [Tit70]) explains the terminology elliptic and hyperbolic.

Theorem 2.8.4 ([Bog08, Ch. 2], [Ser80, I.6.4]). Let 푔 ∈ 퐺 such that 푔 is not an inversion. Then: 80

1. If |푔| = 0, then any vertex in 푇푔 is ﬁxed by 푔. Moreover, let 푃 be any vertex in 푋, and 푄 a

vertex in 푇푔 that is of minimal distance to P. Then 푑(푃, 푄) = 푑(푔(푃 ), 푄) and 푑(푃, 푔(푃 )) = 2 · 푑(푃, 푄).

2. If |푔| > 0, then 푇푔 is isomorphic to the tree 풞∞, an inﬁnite cycle, and 푔 acts on 푇푔 by

translations of length |푔|. Moreover, let 푃 be any vertex in 푋 and 푄 a vertex in 푇푔 of minimal

distance to 푃 . Then the geodesic from 푃 to 푔(푃 ) intersects 푇푔 along the geodesic joining 푄 to 푔(푄), and: 푑(푃, 푔(푃 )) = 2 · 푑(푃, 푄) + |푔|

3. For any 푔, 푕 ∈ 퐺, we have:

(a) For 푛 ∈ ℤ, |푔푛| = |푛| · |푔|.

(b) |푔| = |푕푔푕−1|.

Proof. We ﬁrst prove (1). By deﬁnition, if 푣 ∈ 푉 푇푔 for some elliptic automorphism 푔, then

푑(푣, 푔푣) = 0 which means 푣 = 푔푣. Hence 푇푔 contains all the vertices ﬁxed by 푔.

Suppose 푃 ∈ 푉 푋 and let 푄 be the closest vertex to 푃 that lies in 푇푔. Set 푛 = 푑(푃, 푄). If 푛 = 0, then 푄 = 푃 = 푔(푃 ) and the conclusion is trivial. Now suppose 푛 ≥ 1. Since automorphisms preserve the lengths of geodesics, the length of the geodesic [푃, 푄] is the same as the geodesic [푔(푃 ), 푄]; hence 푑(푃, 푄) = 푑(푔(푃 ), 푄). Now consider the path from 푃 to 푔(푃 ) given by concatenation: [푃, 푄] · [푄, 푔(푃 )]. This is a path from 푃 to 푔(푃 ). Suppose this path were not reduced. Then any backtracking must occur on an edge 푒 that lies on 푄, say 훼(푒) = 푞. But since 푔[푃, 푄] = [푔(푃 ), 푄], we must have 푔휔(푒) = 휔(푒), which contradicts 푄 being of minimal distance to 푃 . Hence, this concatenated path is indeed reduced, and therefore is the geodesic from 푃 to 푔(푃 ). Hence,

푑(푃, 푔(푃 )) = 푑(푃, 푄) + 푑(푄, 푔(푃 )) = 2 · 푑(푃, 푄). 81

To show (2), let |푔| = 푛 and 푣 ∈ 푉 푇푔. Denote the geodesic [v,gv] by the vertices 푣0 =

2 푣, 푣1, 푣2, . . . , 푣푛−1, 푣푛 = 푔푣. If we concatenate the geodesics [푣, 푔푣] and 푔[푣, 푔푣] = [푔푣, 푔 푣], then we have a path from 푣 to 푔2푣. Suppose this path is not reduced. Then a backtracking must occur

at an edge lying on 푔푣, meaning we must have 푣푛−1 = 푔푣1. If 푛 = 1, then this means we have 푣 = 푔2푣 = 푔(푔푣) and thus 푔 would invert an edge, a

contradiction. If 푛 ≥ 2, then we would have 푑(푣1, 푔푣1) = 푛 − 2 < 푛 which contradicts the deﬁnition of translation length. Hence, the geodesic from 푣 to 푔2푣 is just a concatenation of [푣, 푔푣]

and [푔푣, 푔2푣]. It follows by induction that the geodesics 푔푚[푣, 푔푣], 푚 ∈ ℤ form an inﬁnite path 푇 ,

isomorphic to 풞∞ and stable under 푔, with 푔 acting by translations of length |푔| = 푛. Now suppose 푃 is any vertex in 푋 and let 푄 be the vertex in 푇 closest to 푃 . Then the geodesics [푃, 푄], [푄, 푔푄], [푔푄, 푔푃 ] form a reduced path. Thus,

푑(푃, 푔푃 ) = 푑(푃, 푄) + 푑(푄, 푔푄) + 푑(푔푄, 푔푃 ) = 2 · 푑(푃, 푄) + |푔|.

Hence, 푑(푃, 푔푃 ) = |푔| only if 푑(푃, 푄) = 0, so a vertex 푃 ∈ 푇 if and only if 푃 ∈ 푇푔, showing that

푇푔 is isomorphic to 풞∞. Thus (2) is proved. Let 푔 ∈ 퐺. Suppose |푔| = 0. Then by the above, we have that 푔 ﬁxes some vertex 푣 ∈ 푋.

Hence, we clearly have 푔푛(푣) = 푣 for any 푛 ∈ ℤ. So |푔푛| = |푛| · |푔| = 0 in this case. Now suppose |푔| > 0. Applying (2) above, we have for any vertex 푃 that

푑(푃, 푔푛(푃 )) = 2 · 푑(푃, 푄) + |푛| · |푔|,

→ → where 푄 is the closest vertex on 푇푔 to 푃 . Since 푑(푃, 푄) = 0 if and only if 푃 lies on 푇푔, we see

푛 푛 푛 that |푔 | = min푥∈푉 푋 푑(푥, 푔 (푥)) = min → 푑(푥, 푔 (푥)) = |푛| · |푔|. Hence this shows (3)(a). 푥∈푉 푇푔 To show (b), let 푔 and 푕 be in 퐺. Note that 푔 and 푕 are automorphisms, hence are bijective 82 functions on 푋. A simple calculation

|푔| = min 푑(푣, 푔(푣)) 푣∈푋 = min 푑(푕−1(푣), 푔푕−1(푣)) 푕−1(푣)∈푋

= min 푑(푣, 푕푔푕−1(푣)) = |푕푔푕−1| 푣∈푋

shows that |푔| = |푕푔푕−1|.

Now let 푦 ∈ 푕푇푔. Then 푦 = 푕(푣), for some 푣 ∈ 푇푔. Thus,

|푔| = 푑(푣, 푔푣) = 푑(푕−1(푦), 푔(푕−1(푦)) = 푑(푦, 푕푔푕−1(푣)).

−1 Since |푔| = |푕푔푕 |, we must have that 푦 ∈ 푇푕푔푕−1 as well. This shows 푕푇푔 = 푇푕푔푕−1 , whence (c).

Remark 2.8.5. The above is just a special case of the more general situation for the isometry group of a 퐶퐴푇 (0) metric space. See [BH10, Sec. II.6, page 228] for this more general case. In fact, for the 퐶퐴푇 (0) metric space case, there are three types of isometries: elliptic, hyperbolic, and parabolic. The third kind, parabolic, necessarily ﬁx an end of the space. For trees, there is a somewhat similar notion proven by Tits [Tit70], which we state below.

Proposition 2.8.6. Suppose 퐺 is a group acting on a tree 푋 such that each element of 퐺 has a fixed point, but 퐺 has none (so every element of 퐺 can be seen as an elliptic automorphism, but there is no globally fixed point of 푋). Then there exists a unique end of 푋 fixed by 퐺.

Below we state a useful criterion to determine whether or not one has a hyperbolic automorphism. But ﬁrst a deﬁnition.

Deﬁnition 2.8.7. Consider two edges 푒 and 푓 in a tree 푋. Set 푎 = 푑(훼(푒), 훼(푓)) and 푏 = 푑(휔(푒), 휔(푓)). There are three possibilities for 푏 − 푎: 0, 2, or −2. If 푏 − 푎 = 0 we say 푒 and 푓 are coherent and incoherent otherwise. But more precisely, if 푏 − 푎 = −2, we say say 푒 and 푓 are convergently incoherent; if 푏 − 푎 = 2, we say 푒 and 푓 are divergently incoherent. 83 푒 푓

푒 푓

To say it another way, consider the segment [훼(푒), 휔(푓)], the unique geodesic containing both edges. Then 푒 and 푓 are coherent if they are oriented the same way on this segment.

Proposition 2.8.8 ([BL01, Ch. 5], [Bas93, Sec. 6], [Ser80, I.6.4]). Suppose 푒 ∈ 퐸푋 and 푔 ∈ 퐺 such that 푔푒 ̸= 푒. If 푔 is elliptic, then 푒 and 푔푒 are incoherent. In particular, 푒 and 푔푒 are coherent → if and only if 푔 is hyperbolic with 푒 ∈ 퐸푇푔, and |푔| = 푑(훼(푒), 훼(푔푒)).

Proof. Suppose 푔 is an elliptic automorphism. Denote the edge 푒 = (푃, 푄); so 훼(푒) = 푃 and ∘ ′ ′ 휔(푒) = 푄. Let 푄 be the vertex of 푇푔 closest to 푃 . If 푃 is closer to 푄 than 푄, we have 푎 = 푑(푃, 푔푃 ) = 2푑(푃, 푄′)) and 푏 = 푑(푄, 푔푄) = 2 + 2푑(푃, 푄′). So 푏 − 푎 = 2. Clearly, if 푄 is closer to 푄′ than 푃 , then 푏 − 푎 = −2. Hence 푒 and 푔푒 are incoherent. For the second part of the proposition, suppose ﬁrst that 푔 is a hyperbolic automorphism with → 푒 ̸= 푔푒, 푒 ∈ 퐸푇푔, and |푔| = 푑(훼(푒), 훼(푔푒)). Then clearly we have that 푒 and 푔푒 are coherent since both 푃 and 푄 are necessarily translated by |푔| along the axis of 푔. Now suppose that 푒 ̸= 푔푒 are coherent edges. By the above, if 푔 is not hyperbolic, then it must be elliptic, which would contradict 푒 and 푔푒 coherent. So 푒 and 푔푒 coherent for some edge 푒 implies 푔 is hyperbolic. By our coherence assumption, we have that 푑(푃, 푔푃 ) = 푑(푄, 푔푄). → ′ Because 푔 is hyperbolic, there exists a vertex 푄 on 푇푔 closest to one of 푃 or 푄, say WLOG it is 푄. Then 푑(푃, 푔푃 ) = 2 · 푑(푃, 푄′) + |푔| and 푑(푄, 푔푄) = 2 · 푑(푄, 푄′) + |푔|. Hence, we must have → → ′ ′ that 푑(푃, 푄 ) = 푑(푄, 푄 ), which is impossible unless 푃 and 푄 lie on 푇푔. Hence, 푒 ∈ 퐸푇푔 and 푑(훼(푒), 훼(푔푒)) = 푑(푃, 푔푃 ) = |푔|.

→ Corollary 2.8.9. Suppose 푔, 푕 ∈ 퐺 with |푔| > 0 and 푔푒 = 푕푒 for some edge 푒 ∈ 퐸푇푔. Then → |푕| = |푔| > 0, and 푒 ∈ 퐸푇푕. 84 Proof. From the hypothesis we have that 푔 is a hyperbolic automorphism. Since 푒 lies on the axis of 푔, we must have that 푒 ̸= 푔푒. Hence, 푒 and 푔푒 are coherent edges with 푑(훼(푒), 훼(푔푒)) = |푔|. Since 푔푒 = 푕푒, we have that 푒 and 푕푒 are also coherent, and 푑(훼(푒), 훼(푕푒)) = |푔|. So by the above → proposition, 푕 is hyperbolic, 푒 ∈ 푇푕, and |푕| = |푔| > 0.

2.9 Property FA

The main reference for this section is Chapter I, section 6.1 of Serre’s Trees [Ser80]. Let us ﬁrst deﬁne the property that this section is all about:

Definition 2.9.1. Property (FA): A group 퐺 is said to have property (FA) if every action on a tree has a fixed vertex or a fixed geometric edge.

To fix a geometric edge means to fix an edge pair set {푒, 푒}; hence, we could invert an edge, but technically are still fixing a point of the tree. Namely, the middle of the geometric edge. Since it is not too much to ask that a group act without inverting edges, let us just assume for the remainder of these notes that any group action on a tree is without inversion of edges. With that assumption, a group 퐺 having property (FA) must fix some vertex of the tree. The letters “F” and “A” roughly stand for fixed points and tree (because the word tree is “arbre” in French, and Serre - whose is French - first defined property (FA) [Ser80, I.6.1]).

Recall that groups like 퐷∞ and SL2(ℤ) can be written as an amalgamated free product of the

form 퐺1 *퐴 퐺2, with 퐺1 ̸= 퐴 and 퐺2 ̸= 퐴:

∼ ∼ 퐷∞ = 푍2 * 푍2 SL2(ℤ) = 푍4 *푍2 푍6.

∼ Any group 퐺 that can be written in this fashion (퐺 = 퐺1 *퐴 퐺2, 퐺1 ̸= 퐴, 퐺2 ̸= 퐴) will be called an amalgam. The following Theorem gives us an equivalence for property (FA), roughly saying that a group has (FA) if and only if it is not an amalgam.

Theorem 2.9.2. [Ser80, I.6.1, Thm 15] Suppose 퐺 is a countable group. Then 퐺 has property 85 (FA) if and only if the following hold:

1. 퐺 is not an amalgam;

2. 퐺 has no quotient isomorphic to ℤ;

3. 퐺 is ﬁnitely generated.

We sketch a proof similar to that of Serre. But ﬁrst, a simple lemma will help elucidate a part of the argument.

Lemma 2.9.3. Any countable group can be written as the union of an increasing sequence of ﬁnitely generated subgroups.

Proof. Let 퐺 be a countable group. Clearly, if 퐺 is ﬁnitely generated, then we are done. So assume 퐺 is not ﬁnitely generated and let 푆 ⊆ 퐺 be a generating set. Note that every group has some generating set, as we can always take 푆 = 퐺. Since 퐺 is countable and, by assumption, not

finitely generated, 푆 must be countably infinite. Let us write 푆 = {푥1, 푥2, 푥3,...}. Every 푔 ∈ 퐺 can be written as a finite product of elements from 푆; that is, there exists an 푛 ∈ ℕ such that

휀1 휀2 휀푛 푔 = 푥푖(1)푥푖(2) . . . 푥푖(푛),

where 푥푖(푗) ∈ 푆 and 휀푗 = ±1 for 푗 = 1, 2, . . . , 푛. Now we deﬁne inductively an increasing sequence of ﬁnitely generated subgroups:

퐺1 = < 푥1 >

퐺2 = < 푥1, 푥2 >

퐺3 = < 푥1, 푥2, 푥3 > . .

퐺푘 = < 푥1, . . . , 푥푘 > . . 86

Clearly we have 퐺1 ⊂ 퐺2 ⊂ ..., all of which are ﬁnitely generated subgroups of 퐺 by deﬁnition.

Note that, for every 푘, 퐺푘 ̸= 퐺, as 퐺 is not ﬁnitely generated. ⋃︀∞ Now we claim that 푘=1 퐺푘 = 퐺. One inclusion is obvious, and for any 푔 ∈ 퐺

휀1 휀2 휀푛 푔 = 푥푖(1)푥푖(2) . . . 푥푖(푛), 푛 ≥ 1,

as noted above. Each 푥푗 lies in 퐺푗, hence 푔 ∈ 퐺푀 where 푀 = max푗=1,...,푛 푖(푗). Thus 퐺 = ⋃︀∞ 푘=1 퐺푘 as claimed.

Remark 2.9.4. The hypothesis in Theorem 2.9.2 that 퐺 be countable can be dropped [Ser80, I.6.1], but one must replace condition (3) with the following:

(3’). 퐺 is not the union of a strictly increasing sequence of subgroups.

Proof of Theorem 2.9.2. Suppose our countable group 퐺 has property (FA).

[(FA) =⇒ (1):] If 퐺 is an amalgam, then this means we can write 퐺 = 퐴 *퐶 퐵, where 퐴 ̸= 퐶 ̸= 퐵. From Bass-Serre theory, this means there exists a tree 푇 on which 퐺 acts with fundamental domain a segment, given by the edge pair {푒, 푒}, with vertices 훼(푒) = 푋 and 휔(푒) =

푌 . In particular, 퐺푋 = 퐴, 퐺푌 = 퐵, and 퐺푒 = 퐶. The vertices of the tree can be identiﬁed with left cosets of 퐴 and 퐵 in 퐺, and edges left cosets of 퐶 in 퐺. Since 퐺 is distinct from 퐴 and 퐵 (recall we can write every element in 퐺 in a “normal form,” a product of elements that alternate between 퐴 and 퐵), we have that 퐺 cannot possibly ﬁx any vertex of 푇 . This contradicts 퐺 having property (FA), hence 퐺 cannot be an amalgam. ∼ [(FA) =⇒ (2):] Suppose 퐺 contains a normal subgroup 푁 such that 퐺/푁 = ℤ. We choose a generating element 푡 = 푡푁 in 퐺/푁. Then for every 푔 ∈ 퐺, we have 푔 = 푔푁 = (푡)푛, for some

푛 ∈ ℤ. Thus, we can deﬁne an action of 퐺 on the inﬁnite straight path 풞∞ :

풞∞ 푒−1 푒0 푒1 -1 0 1 2

푛 Each 푔 ∈ 퐺 acts on 풞∞ by translation. That is, 푔 = (푡) for some 푛 ∈ ℤ, and we deﬁne 87

퐺 ↷ 풞∞ by 푔 · 푒푖 ↦→ 푒푖+푛. Clearly this defines an action that is fixed point free (i.e. no point is fixed by every element of 퐺), and this contradicts property (FA). [(FA) =⇒ (3):] By Lemma 2.9.3, 퐺 is the union of an increasing sequence of finitely gener-

ated subgroups, 퐺1 ⊂ 퐺2 ⊂ . . . 퐺푘 ⊂ ... . We form a graph 푋 by taking the disjoint union of all

left cosets 퐺/퐺푛 as vertices: ∞ ∞ ᨁ ᨁ V 푋 = 퐺/퐺푛 = 푔퐺푛 푛=1 푛=1, 푔∈퐺 We deﬁne the edge set E 푋 by declaring two vertices adjacent if and only if they belong to consecutive sets 퐺/퐺푛 and 퐺/퐺푛+1, and they correspond under the canonical map

퐺/퐺푛 → 퐺/퐺푛+1, given by [푔]푛 ↦→ [푔]푛+1, where here [푔]푘 denotes equivalence modulo 퐺푘. Now I claim that 푋 is a tree. First we show 푋 is connected. The best way to visualize 푋 is to start with the trivial cosets 퐺1, 퐺2, . . . , 퐺푘,..., which obviously form an inﬁnite path in the tree beginning at 퐺1. Think of this as the “spine” of the tree. For any 퐺푖 along our spine, we have that

퐺푖 is adjacent to all cosets of the form 푔퐺푖−1, where 푔 ∈ 퐺푖.

Now if we think of all cosets in 퐺/퐺푛 being at the same “level,” we can schematically draw these cosets above and below the vertex 퐺푛. Every coset at any level (that isn’t the spine) is connected to a single coset in the next (plus one) level. If we follow these paths starting at any coset, we will eventually reach the “spine” at some 퐺푀 .

To show 푋 is connected, It sufﬁces to show that there is a path from any coset 푔퐺푘 to 퐺1. So ⋃︀∞ let 푔퐺푘 be any vertex. Since 퐺 = 푛=1 퐺푛, we have that there exists a natural number 푀 such that 푔 ∈ 퐺푀 . Hence, we have a path

푔퐺푘, 푔퐺푘+1, 푔퐺푘+2, . . . , 푔퐺푀−1, 푔퐺푀 = 퐺푀

from 푔퐺푘 to our “spine” path. Hence, we have a path from 푔퐺푘 to 퐺1 and so 푋 must be connected.

Furthermore, since all cosets in 퐺/퐺푛 are disjoint for each 푛, 푋 does not have any loops. Thus 푋 is indeed a tree. Now we use the hypothesis that 퐺 has property (FA). 퐺 acts in a natural way on the tree 88 (permutes cosets at the same “level”) and by property (FA), we have that there exists a ﬁxed vertex, say 푔퐺푛 for some 푛. This means, ∀푕 ∈ 퐺, 푕 · (푔퐺푛) = (푕푔)퐺푛 = 푔퐺푛. This implies that

−1 푔 푕푔 ∈ 퐺푛 for all 푕 ∈ 퐺. So in particular, 푔 ∈ 퐺푛 (let 푕 = 푔) and thus 푕 ∈ 퐺푛 for all 푕 ∈ 퐺.

Hence, we must have 퐺 = 퐺푛 and so 퐺 is finitely generated. Now we may show that converse: suppose a countable group 퐺 satisfies (1), (2), and (3); we must show 퐺 acting on any tree has a fixed point. So let 푋 be any tree and assume 퐺 acts on 푋. Let 푌 = 퐺∖푋, the quotient graph of 푋 by the action of 퐺. We know by the Structure Theorem (Theorem 2.4.1, Theorem 2.4.2) for groups acting on trees that, if we choose a maximal subtree 푇 in 푌 and a lift of this in 푋, then 퐺 will be isomorphic to the fundamental group of a certain graph of groups, built from the stabilizers of lifts of vertices and edges in 푌 . However, before we start with that, let us examine the structure of 푌 .

Recall that, if (ℍ, 푍) is a graph of groups, then 휋1(ℍ, 푍, 푇 ) denotes the fundamental group of this graph of groups, which is generated by the vertex groups and edge groups of (ℍ, 푍), with some relations (the 푇 is a maximal subtree of the graph 푍). If we let 푅 denote the normal subgroup

of 휋1(ℍ, 푍, 푇 ) generated by all the vertex groups, then we saw in Example 2.3.13 that we must ∼ have 휋1(ℍ, 푍, 푇 )/푅 = 휋1(푍).

In particular for our case here, we have that 휋1(푌 ) is isomorphic to such a quotient of 퐺, cf.

[Ser80, Corollary 1, I.5.4]. Since 휋1(푌 ) is a free group (the fundamental group of any graph is ∼ free), it is isomorphic to 퐹푛 = ℤ * ℤ *···* ℤ (n-times) for some 푛. If 푛 > 0, then we would have that 퐺 contains a quotient isomorphic to ℤ (since we could just quotient further and apply the 3rd

and 4th isomorphism theorems). But this is a contradiction to (2), hence we must have that 휋1(푌 ) is trivial. This means that 푌 , the quotient graph, is a tree.

Now let 푌̃︀ be a lift of 푌 in 푋 (which we know exists by Proposition 2.2.2). Using this lift, we

can deﬁne a graph of groups (픾, 푌 ) with

∼ 퐺 = 휋1(픾) = ⟨풢푣, 풢푒 | 휙푒(푎) = 휙푒(푎), 푎 ∈ 풢푒⟩ 89 where 풢푣 = Stab퐺(푣̃︀) and 풢푒 = Stab퐺(푒̃︀), with 푣 and 푒 vertices and edges in 푌 and 푣̃︀ and 푒̃︀ represent their respective lifts in 푌̃︀. Because a graph of groups is built up from amalgams of vertex

′ ′ and edge stabilizers, we can also write 퐺 as the union of groups 휋1(픾, 푌 ), where 푌 is a ﬁnite subtree of 푌 :

⋃︁ ′ 퐺 = 휋1(픾, 푌 ) 푌 ′⊂푌 푌 ′ﬁnite subtree

′ ′ Because 퐺 is finitely generated, we must have that 퐺 = 휋1(픾, 푌 ) for some finite subtree 푌 . There could be several finite subtrees that yield this equality (any 푌 ′′ ⊃ 푌 ′ would work), so let us choose a minimal one and also call it 푌 ′ (minimal in the sense that, if we took a vertex or edge

′ away, then we would no longer have 퐺 = 휋1(픾, 푌 )). Now we could have that 푌 ′ is just a single vertex. In that case, we have that 퐺 equals the stabilizer of that vertex (really, a lift of the vertex of 푌 ′ to 푋), and hence we have a ﬁxed point in 푋. If 푌 ′ is not just a lone vertex, then since it is ﬁnite, there must be a terminal vertex, say 푧. One can check that 푌 ′ is a tree if and only if 푌 ′′ = 푌 ′ ∖ {푧} is a tree. If 푒 denotes the edge from 푧 in

′′ ′ ′ 푌 into 푌 , then by the deﬁnition of 휋1(픾, 푌 ) we must have

′ ′′ 퐺 = 휋1(픾, 푌 ) = 휋1(픾, 푌 ) *퐺푒 퐺푧.

′ ′′ Because of the minimality of 푌 , we have 퐺 ̸= 휋1(픾, 푌 ) and, even more so, 퐺 ̸= 퐺푧. But then 퐺 is an amalgam and this contradicts (1). Hence, we must have that 푌 ′ is a single vertex, and therefore 퐺 ﬁxes a point in 푋 (the lift of the single vertex in 푌 ′). Since 푋 was arbitrary, we have that 퐺 has property (FA).

Examples

1. SL3(ℤ) has property (FA).

Serre proves this in [Ser80, I.6.6. Theorem 16].

2. More generally, for 푛 ≥ 3 one has that SL푛(ℤ) and GL푛(ℤ) have property (FA). In fact any group with Kazdhan’s property (T) has property (FA) [Wat82]. 90

3. [Bog08, Thm 13.10] For 푛 ≥ 3 the automorphism group Aut(퐹푛) of the free group of rank 푛 has property (FA).

4. Any ﬁnitely generated torsion group has (FA).

This is decently easy to check. Such a group already satisﬁes (2) and (3) of the Theorem. Such a group can’t be an amalgam, since amalgams necessarily have elements of inﬁnite order.

5. If a subgroup 퐻 ≤ 퐺 of ﬁnite index has (FA), then so does 퐺. More precisely,

Lemma 2.9.5. Suppose 퐺 is a group acting on a tree 푋 without inversions, and let 퐻 be a subgroup of ﬁnite index in 퐺. If 퐻 has (FA), then so does 퐺.

⋂︀ −1 Proof. Take 푁 = 푔∈퐺 푔퐻푔 . This is a ﬁnite index normal subgroup of 퐺 contained in 퐻.

The quotient group 퐺/푁 naturally acts on the tree of 퐻-fixed points 푋퐻 . Since 푁 is finite index, 퐺/푁 is a finite group, and hence has bounded orbit in its action on 푋퐻 . Thus 퐺/푁

has a ﬁxed point in 푋퐻 , and hence 푋퐺 ̸= ∅ also.

6. The converse of the above is not true: If a group 퐺 has (FA), then it is not necessarily true that every ﬁnite index subgroup has property (FA). Let 퐺 be the group:

퐺 = ⟨︀푥, 푦 | 푥푎 = 푦푏 = (푥푦)푐⟩︀

where 푎, 푏, 푐 ≥ 2 are integers. Such a 퐺 has (FA) since each of 푥, 푦 and 푥푦 are of finite order, hence have fixed points; moreover, there is a fixed point common to both 푥 and 푦, and hence fixed by all of 퐺 since 퐺 is generated by these two elements. Now if

1 1 1 + + ≤ 1, 푎 푏 푐

then 퐺 contains a ﬁnite index subgroup isomorphic to 휋1(Σ푔), 푔 ≥ 1. That is, the fundamental group of a compact oriented surface of genus 푔. Such a group has a quotient isomorphic 91 to ℤ, so by the above theorem, does not have (FA).

We can give another example of a ﬁnite index subgroup that does not have property (FA) in a group that does, but ﬁrst we need another result from Serre, which we generalize in the next section.

2.10 Nilpotent Groups Acting on a Tree

Recall that a group is called nilpotent if one of the terms in its lower central series equals {1}. In a sense, they are the groups that are “almost” abelian. Every nilpotent group 퐺 is solvable, meaning there exists a normal series

{1} = 퐻0 ⊴ 퐻1 ⊴ ··· ⊴ 퐻푛 = 퐺

where each successive quotient 퐻푖+1/퐻푖 is abelian. A slightly smaller class of groups is deﬁned in a similar fashion:

Deﬁnition 2.10.1. A group 퐺 is called polycyclic if there exists a subnormal series

{1} = 퐺0 ⊲ 퐺1 ⊲ ··· ⊲ 퐺푛 = 퐺

such that 퐺푖+1/퐺푖 is cyclic for all 푖 ≥ 0. We call such a series a cyclic series. The length is the number of non-trivial groups in the sequence.

If each quotient 퐺푖+1/퐺푖 is infinite cyclic, then we call 퐺 poly-풞∞, or strongly polycyclic. We also declare the trivial group to be poly-풞∞. If 퐺 is polycyclic, we define the Hirsch number or Hirsch length of 퐺 to be the number of infinite cyclic factors on a cyclic series of 퐺. This number is the same for any cyclic series of 퐺 and hence is an invariant of the group [DK, prop. 11.3].

Remark 2.10.2. Let us collect here some facts concerning polycyclic groups. 92 ∙ Every polycyclic group is solvable.

Since every cyclic group is abelian, this is straightforward from the deﬁnition of polycyclic and solvable.

∙ Polycyclic groups are finitely generated; a polycyclic torsion group (all elements have finite order) is finite; quotients of polycyclic groups are polycyclic; subgroups of polycyclic groups are polycyclic. [DK, prop 11.3].

∙ If 푁 ⊲ 퐺, 푁 is polycyclic and 퐺/푁 is polycyclic, then 퐺 is polycyclic [DK, prop 11.3].

∙ Every ﬁnitely generated nilpotent group is polycyclic [DK, Prop. 11.5].

∙ [DK, 11.8] Every polycyclic group contains a normal subgroup of ﬁnite index which is poly-

풞∞.

푛 푚 푚 Example 2.10.3. A proto-typical polycyclic group is given by 퐺 = ℤ ⋊휙 ℤ , where 휙 : ℤ → Aut(ℤ푛) is a homorphism giving the action of ℤ푚 on ℤ푛. Clearly ℤ푛 is polycyclic, as is the quotient 퐺/ℤ푛. Hence by (3) in the above remark, 퐺 is polycyclic (in fact 퐺 is strongly polycyclic).

Example 2.10.4. The inﬁnite dihedral group can be written as a semi-direct product 퐷∞ = ℤ ⋊ ℤ/2ℤ, hence has a normal series:

{1} ⊲ ℤ ⊲ ℤ ⋊ ℤ/2ℤ.

The successive quotients are ℤ and ℤ/2ℤ, thus 퐷∞ is a polycyclic group. However, it is not strongly polycyclic. One can show [DK, Thm 10.47] that the collection of finite order elements Tor(퐺) in a nilpotent group 퐺 forms a subgroup. Moreover, if 퐺 is finitely generated, then Tor(퐺) must be finite.

In 퐷∞, we know that the automorphisms 푎 and 푐 (cf. Example 2.1.4) generate the group and are of ﬁnite order 2. But, their product is the automorphism 푏, which we know has inﬁnite order.

This shows that the collection of ﬁnite order elements Tor(퐷∞) cannot be a subgroup, and hence

퐷∞ is not nilpotent. 93 Above we have described the two types of automorphisms that a tree can have (if we ignore inversions; otherwise there are three!). Hence groups that act on a tree act by either elliptic or hyperbolic elements of Aut(푋). As we described in the previous section, groups that have a global fixed point are said to have property (FA). So necessarily, such groups act only by elliptic automorphisms, since no hyperbolic automorphism can fix a point. For what other groups that act on a tree can we say whether or not the action has a fixed point or not? Perhaps a better question is, which groups 퐺 act on a tree 푋 by only elliptic or only hyperbolic elements? By the following theorem of Serre, we know that for finitely generated nilpotent groups this is the case:

Theorem 2.10.5 ([Ser80, I.6.5, Proposition 27]). Let 퐺 be a ﬁnitely generated nilpotent group acting on a tree 푋. Then there are two mutually exclusive cases:

1. 퐺 has a ﬁxed point.

2. There is a straight path 푇 invariant under 퐺 on which 퐺 acts by translations via a non-trivial

homomorphism 퐺 → ℤ.

Proof. From Theorem 2.8.4 we know that, if an element 푔 ∈ 퐺 is hyperbolic, then it ﬁxes no point of 푋. Hence the two cases must be mutually exclusive. Since any ﬁnitely generated nilpotent group is polycyclic, we can choose a normal series of some length 푛:

{1} = 퐺0 ⊴ 퐺1 ⊴ ··· ⊴ 퐺푛 = 퐺

where each quotient 퐺푖+1/퐺푖 is cyclic. We show that any such 퐺 satisﬁes 1) or 2) by induction on 푛. For the base case we have that 푛 = 0, which means 퐺 is a trivial group and hence ﬁxes all of 푋.

Now suppose the result holds for 푛 − 1 where 푛 ≥ 1, and let 퐻 = 퐺푛−1. By the inductive hypothesis, we have 퐻 satisfies 1) or 2). 94 Suppose that 퐻 has a fixed point and denote the fixed subtree of 푋 under the action of 퐻 by 푋퐻 . The quotient group 퐺/퐻 is cyclic, and naturally acts on the tree 푋퐻 (for 푥 ∈ 푋퐻 ,

푔퐻 · 푥 = 푔푥). By prop. 26 of [Ser80, I.6.5], we have that 퐺/퐻 has a ﬁxed point in 푋퐻 ; so there

exists some point 푥0 ∈ 푋퐻 such that 푔퐻 · 푥 = 푥 for all 푔 ∈ 퐺. Hence, 푔푥 = 푥 for all 푔 ∈ 퐺 and so 퐺 has a ﬁxed point. Now suppose that 퐻 has no ﬁxed point. Thus 퐻 acts on 푇 solely by hyperbolic automorphisms and by the hypothesis, there is an axis 푇 for 퐻, with 퐻푇 = 푇 , and 퐻 acts by non-trivial translations on 푇 . By (3)(c) of Theorem 2.8.4, since 퐻 is normal in 퐺, we have that 퐺푇 = 푇 as well,

since 푔푇푕 = 푇푔푕푔−1 for all 푔 ∈ 퐺 and all 푕 ∈ 퐻, and the subtree 푇 is the intersection of all the 푇푕 ∼ subtrees. Therefore, the action of 퐺 on 푋 induces a homomorphism 휙 : 퐺 → Aut(푇 ) = 퐷∞.

Since we have a non-trivial homomorphism 퐻 → ℤ, we have that Im(휙) contains a non-trivial group of translations. The only possibility then for Im(휙) is 퐷∞ or ℤ. If Im(휙) = 퐷∞, then ∼ 퐺/ ker(휙) = 퐷∞ (by ﬁrst isomorphism theorem); but quotients of nilpotent groups are nilpotent, ∼ and 퐷∞ is not nilpotent. Hence, the only possibility is for Im(휙) = ℤ, which is case (2).

Notice that there were only two places in the above proof where nilpotency was utilized: before applying induction on the length of the normal series, we deduced the existence of one using the fact that ﬁnitely generated nilpotent groups are polycyclic; and lastly, we used the nilpotency of 퐺

to rule out a quotient isomorphic to 퐷∞. One could perhaps hope of generalizing Theorem 2.10.5 by replacing 퐺 with, say, a polycyclic group instead. But as we saw in Example 2.10.4, the inﬁnite dihedral group is polycyclic, and its presence as the automorphism group of a straight path is why the theorem is stated for nilpotent groups. One might then try to assume that 퐺 is simply strongly polycyclic instead. But there, too, we

have a problem: although 퐷∞ is not strongly polycyclic, quotients of strongly polycyclic groups need not be strongly polycyclic. Hence, the same proof cannot be applied in this case. Sadly we cannot say more about this attempt to generalize the above theorem at this time. What started out as a simple thought turned into (for the author) a wild journey into a lot of very 95 exciting literature. Most notably, the book by Segal [Seg83] that was the ﬁrst deﬁnitive book on polycyclic groups. Several chapters are dedicated to showing a classical result of Auslander that every polycyclic group 퐺 can be canonically embedded into GL푛(ℤ). Perhaps more can be said concerning polycyclic groups with property (FA), but we shall not be able to address that here. Lastly, we note that there has been more recent research concerning property (FA) and some generalizations. See for instance [CV96], where a group theoretic criterion on the generators of a group determines if it has property (FA). See also [Far09] where a generalization of property (FA) is given. 96

CHAPTER 3 TOPOLOGY OF AUT(X) AND LATTICES

In this chapter we study closely the topological nature of the automorphism group of a locally finite tree. From this we discuss tree lattices in the automorphism group. These are discrete subgroups that have finite volume (in the sense of Definition 2.3.14 and Definition 2.7.4). We mention some important results on the existence and structure of tree lattices from [BL01]. The interested reader can look there and at the many additional references found within for more on tree lattices.

3.1 Topological Preliminaries

In this section, we record a few basic results concerning topological groups. Firstly, recall the following deﬁnitions from topology:

Deﬁnition 3.1.1. Let 푌 be a topological space.

푌 is said to be Hausdorff or satisﬁes the 푇2 axiom if, for any two distinct points 푥, 푦 ∈ 푌 , there exists disjoint open sets 푈 and 푉 containing 푥 and 푦, respectively. We say 푌 satisﬁes the 푇0 axiom if, for any two distinct points 푥 and 푦, there exists an open set 푈 containing one of 푥 or 푦, but not the other. ⋃︀ 푌 is said to be compact if, for every collection {푈훼}훼∈퐼 of open sets in 푌 with 푌 ⊆ 훼 푈훼

(called an open cover of 푌 ), there necessarily exists a finite subcollection {푈훼푖 }훼푖 , 푖 = 1, 2, . . . , 푛 ⋃︀푛 such that 푌 ⊆ 푖=1 푈훼푖 (this is called a finite sub-cover). We say 푌 is locally compact if at every point 푦 ∈ 푌 there exists a compact neighborhood of 푦. If a space 푌 is Hausdorff, then one can equivalently define locally compact to mean for every point 푦, every neighborhood 푈 of 푦 contains a compact neighborhood; i.e. there exists compact 퐾 with 퐾 ⊂ 푈. A separation of 푌 is a pair (푈, 푉 ) of non-empty disjoint open sets such that 푌 = 푈 ∪ 푉 . We say 푌 is connected if no separation of 푌 exists. We say that the space 푌 is totally disconnected if the only connected subspaces of 푌 are 97 singletons.

Deﬁnition 3.1.2. A group 퐺 is said to be a topological group if 퐺 is endowed with a topology such that the group operations are continuous with respect to the topology. Hence, the maps (푔, 푕) → 푔푕 from 퐺 × 퐺 to 퐺, and 푔 → 푔−1 from 퐺 to 퐺 are continuous (here 퐺 × 퐺 is given the product topology induced by the topology on 퐺).

As a simple consequence of the deﬁnition, we have that for any 푔, 푕 ∈ 퐺 and open neighborhood 푈 of 푔푕, there exists neighborhoods 푉푔 and 푉푕 of 푔 and 푕, respectively, such that 푉푔푉푕 ⊂ 푈; similarly, for any neighborhood 푈 of 푔−1 there exists a neighborhood 푊 with 푊 ⊂ 푈. The following proposition contains basic results on topological groups. See the given reference(s) for details.

Proposition 3.1.3. Let 퐺 be a topological group.

1. [Pon39, III.16] For 푎 a ﬁxed element of 퐺, the functions 퐿푎, 푅푎, and 휙 from 퐺 to 퐺 deﬁned by

−1 퐿푎(푔) = 푎푔 푅푎(푔) = 푔푎 휙(푥) = 푥 , 푥 ∈ 퐺,

are homeomorphisms. Consequently, 퐺 is a homogeneous space (퐺 acts transitively by homeomorphisms on itself).

2. [Pon39, III.16], [Fol95, Prop. 2.1] Let 푈 be open and 퐹 closed in 퐺. For any point 푔 ∈ 퐺 and any subset 퐴 ⊂ 퐺, the sets 푔퐹, 퐹 푔, and 퐹 −1 are closed and the sets 퐴푈, 푈퐴, 푈 −1 are open in 퐺.

3. [Fol95, Prop 2.1] For any open neighborhood 푈 of the identity element id ∈ 퐺, there exists

a symmetric neighborhood 푉 of id such that 푉 푉 −1 = 푉 푉 = 푉 2 ⊂ 푈 (a symmetric neighborhood 푉 means 푉 = 푉 −1).

4. [Fol95, Prop 2.1] If 퐻 ≤ 퐺, then the closure of 퐻 in 퐺, written 퐻, is also a subgroup.

5. [Fol95, Prop 2.1] If 푈 ≤ 퐺 is open, then 푈 is also closed. 98

6. [HR79, (4.8), (8.4)], [MZ55, 1.18], [Dik13, Ch.4] 퐺 is 푇0 ⇐⇒ 퐺 is 푇1 ⇐⇒ 퐺 is

푇2 ⇐⇒ 퐺 is 푇3 ⇐⇒ 퐺 is 푇 1 . 3 2

Here are two more facts concerning locally compact groups.

Proposition 3.1.4 ([HR79, (5.22)]). Let 퐺 be a locally compact topological group and 퐾 a subgroup of 퐺. Then the quotient space 푋 = 퐺/퐾 is locally compact.

Proposition 3.1.5 ([Bou71, III.2.5, Prop 13]). Let 퐺 be a topological group, 퐾 a subgroup of 퐺. Then the quotient space 푋 = 퐺/퐾 is Hausdorff if and only if 퐾 is closed.

Note that we are assuming no separation axioms on 퐺; it could be a space that is not even 푇0. Lastly, we mention a simple result on discrete subgroups.

Deﬁnition 3.1.6. A topological group 퐺 is said to be discrete if, for any 푔 ∈ 퐺 there exists a neighborhood 푈 of 푔 such that 푈 = {푔}.

Proposition 3.1.7 ([HR79, (5.11)]). Let 퐺 be a topological group, Γ ≤ 퐺 a discrete subgroup. Then Γ is closed in 퐺.

3.2 Topology of Aut(푋)

Let 푋 be a locally ﬁnite tree and 퐺 = Aut(푋). There are a few ways that one can endow 퐺 with a topology to make it a topological group. That is, having the group operation continuous with respect to this topology. One such topology is given by the following subbasis elements, which say that in this topology two automorphisms are deemed “close” if they agree on a ﬁnite subtree of 푋:

풮(푓, 푇 ) = {푔 ∈ 퐺 | 푓|푇 = 푔|푇 }

where 푓 ∈ 퐺 and 푇 is a ﬁnite subtree of 푋. Clearly the collection of all such sets 풮(푓, 푇 ) forms a subbasis, since for all 푓 ∈ 퐺, we have 푓 ∈ 풮(푓, 푇 ) for any ﬁnite subtree 푇 . Hence

⋃︁ 퐺 = 풮(푓, 푇 ) 99 where the union is taken over all automorphisms 푓 ∈ 퐺 and 푇 is any ﬁnite subtree in 푋. The main theorem of this section is the following:

Theorem 3.2.1. Suppose 푋 is a locally ﬁnite tree. Then 퐺 = Aut(푋) is a locally compact, Hausdorff, totally disconnected group.

Proof. The proof of this theorem will be collected from a series of lemmas: 퐺 is a topological group Lemma 3.2.2; 퐺 is Hausdorff Lemma 3.2.3; 퐺 is locally compact Lemma 3.2.9; 퐺 is totally disconnected Lemma 3.2.11.

Now for the series of lemmas. The ﬁrst shows that 퐺 is indeed a topological group.

Lemma 3.2.2. 퐺 = Aut(푋) endowed with the topology generated by subbasis elements

풮(푓, 푇 ) = {푔 ∈ 퐺| 푓|푇 = 푔|푇 } is a topological group.

Proof. We need to show that the group operations of 퐺 are continuous with respect to the topology. So we need that the maps

퐺 × 퐺 −→푚 퐺 퐺 −→퐼푛푣 퐺

(푓, 푔) ↦→ 푓푔 푓 ↦→ 푓 −1 are continuous. It sufﬁces to show that the inverse images of the subbasis elements 풮(푓, 푇 ) are open under these maps to prove continuity. So consider 풮(푓, 푇 ) for 푓 ∈ 퐺 and 푇 ⊂ 푋 a ﬁnite subtree. Set

푊 = 푚−1 (풮(푓, 푇 )) = {(푔, 푕) ∈ 퐺 × 퐺| 푔푕 ∈ 풮(푓, 푇 )}

That is, (푔, 푕) ∈ 푊 if 푔|푕(푇 ) = 푓|푇 . Note that 푕(푇 ) is also a subtree of 푋 because 푕 is a bijective graph morphism. We need to show that 푊 is open in 퐺 × 퐺. The group 퐺 × 퐺 is naturally given the product topology. Thus, a set of the form 풮(푓, 푇 ) × 풮(푔, 푇 ′) will be open in 퐺 × 퐺, since each component 100 set is open in 퐺. Now let (푔, 푕) ∈ 푊 . Deﬁne

푉푔,푕 = 풮(푔, 푕(푇 )) × 풮(푕, 푇 )

′ ′ ′ ′ This is an open set in 퐺 × 퐺, and if (푔 , 푕 ) ∈ 푉푔,푕, then 푔 |푕(푇 ) = 푔|푕(푇 ) and 푕 |푇 = 푕|푇 . Thus we

′ ′ have that 푔 |푕′(푇 ) = 푔 |푕(푇 ) = 푔|푕(푇 ) ∈ 푊 . Thus 푉푔,푕 ⊂ 푊 and we may conclude that 푊 is open in 퐺 × 퐺. Therefore multiplication is continuous. To show that the inverse map is continuous, let 풮(푓, 푇 ) be a basis element. Then,

−1 −1 퐼푛푣 (풮(푓, 푇 )) = {푔 ∈ 퐺| 푔 |푇 = 푓|푇 }.

So for all 푥 ∈ 푇, 푔−1(푥) = 푓(푥). This holds if and only if 푓 −1(푦) = 푔(푦) for all 푦 ∈ 푓(푇 ). Thus,

퐼푛푣−1 (풮(푓, 푇 )) = 풮(푓 −1, 푓(푇 )),

which is an open set in 퐺. Therefore the inverse operation is continuous and we may conclude that 퐺 is a topological group.

Recall that the 푇0 axiom for a topological space means, for any two distinct points, there exists

an open set containing one but not the other. If a topological group is at least 푇0, then it is also

regular (and is even 푇 1 ; Proposition 3.1.3). We can easily see that 퐺 is 푇0. 3 2

Lemma 3.2.3. 퐺 = Aut(푋) equipped with the topology above is 푇0, and therefore is Hausdorff, regular, and completely regular.

Proof. Let 푓 and 푔 be distinct automorphisms of 푋. Then there exists a vertex 푥 ∈ 푉 푋 such that 푓(푥) ̸= 푔(푥). The basis element 풮(푓, {푥}) (the ﬁnite subtree consisting of just the single vertex 푥) clearly contains 푓, but does not contain 푔.

A space is called completely regular if it is 푇 1 and as noted above this is equivalent to 푇0 for 3 2 topological groups. Since any space that is completely regular is also regular and regular spaces are also Hausdorff, we can conclude that 퐺 has all three separation properties. 101

Let 퐵푟(푥) denote the ball of radius 푟 in 푋 (recall 푋 is naturally a metric space using the lengths of geodesics between vertices):

퐵푟(푥) = {푦 ∈ 푋| 푑(푥, 푦) < 푟}

Because of the local ﬁniteness of 푋, a ball 퐵푟(푥) will be a ﬁnite subtree in 푋. For 푥 ∈ 푉 푋, let

퐺푥 denote the vertex stabilizer:

퐺푥 = {푔 ∈ 퐺| 푔푥 = 푥}

Notice that this is the same thing as one of our subbasis elements, namely 퐺푥 = 풮(1퐺, 푇푥), where

0 1 1퐺 is the identity automorphism and 푇푥 is a trivial tree with 푇 = {푥} and 푇 = ∅. So vertex stabilizers 퐺푥 are open subgroups in 퐺, and moreover are open symmetric neighborhoods of the identity. We in fact could deﬁne the topology on 퐺 = Aut(푋) another way, via a symmetric neighborhood basis formed by the vertex stabilizers 퐺푥.

Below we show that each vertex stabilizer 퐺푥 is a profinite group: an inverse limit of an inverse system of discrete finite groups. We first define the terms inverse limit and inverse system (cf. [RZ10]).

Deﬁnition 3.2.4. [RZ10, Ch. 1] Suppose (퐺푖)푖∈퐼 is a family of topological groups, where 퐼 is a partially ordered indexing set. We denote the partial ordering with ≤. Suppose further that there is a collection of continuous homomorphisms {휙푖,푗 : 퐺푗 → 퐺푖, 푖 ≤ 푗} satisfying the following conditions:

1. 휙푖,푖 = id퐺푖 , the identity morphism on 퐺푖.

2. 휙푖,푘 = 휙푖,푗 ∘ 휙푗,푘, for all 푖 ≤ 푗 ≤ 푘.

If a family of groups with continuous homomorphisms satisﬁes these conditions, then it is called an inverse system and is denoted {퐺푖, 휙푖,푗, 퐼}. If the mappings 휙푖,푗 are surjective for all 푖 ≤ 푗, then we call {퐺푖, 휙푖,푗, 퐼} a surjective inverse system. 102 Given an inverse system, one can deﬁne its inverse limit 퐺 as a subgroup of the direct product of 퐺 ’s: 푖 {︃ ⃒ }︃ ∏︁ ⃒ 퐺 = lim 퐺 = (푎 ) ∈ 퐺 ⃒ 푎 = 휙 (푎 ) ∀푖 ≤ 푗 ←− 푖 푖 푖⃒ 푖 푖,푗 푗 푖∈퐼 푖∈퐼 ⃒

Since 퐺 is a subgroup of the product, we have natural projection maps 휋푖 : 퐺 → 퐺푖 that pick out

the 푖-th coordinate of an element in 퐺; precisely, 휋푖 is the restriction of the canonical projection ∏︀ map 푖∈퐼 퐺푖 → 퐺푖 . The maps 휋푖 are continuous group homomorphisms, but may not be surjec-

tive. They satisfy the condition 휋푖 = 휙푖,푗 ∘휋푗 for all 푖 ≤ 푗; that is, the following diagram commutes for all 푖 ≤ 푗: 퐺 휋푖 휋푗 퐺 퐺 푗 휙푖,푗 푖

We write (퐺, 휋푖) for the inverse limit (by abuse of notation). This limit satisﬁes the following universal property:

If (퐻, 휓푖) is another inverse limit for the inverse system {퐺푖, 휙푖,푗, 퐼}, then there exists

a unique continuous homomorphism Ψ: 퐻 → 퐺 such that 휋푖 ∘ Ψ = 휓푖 for all 푖 ∈ 퐼. That is, all triangles commute in the diagram below: 퐻

Ψ 휓푖

퐺

휓푗 휋푗 휋푖 퐺 퐺 푗 휙푖,푗 푖 By a standard argument, one can show that inverse limits are unique up to topological isomor-

phism [RZ10, Ch. 1, Prop 1.1.1]. One can also show that, if {퐺푖, 휙푖,푗, 퐼} is a surjective inverse system, then the inverse limit (퐺푖, 휋푖) has the mappings 휋푖 surjective for all 푖 ∈ 퐼 [RZ10, Prop. 1.1.10]. One can generally assume that a given inverse system is surjective by the following:

Proposition 3.2.5 ([RZ10, Corollary 1.1.8]). Suppose {퐺푖, 휙푖,푗, 퐼} is an inverse system, but is not surjective and let

퐺 = (퐺 , 휋 ) = lim 퐺 푖 푖 ←− 푖 푖∈퐼 103 ′ ′ be the inverse limit. Let 휙푖,푗 be the restriction of 휙푖,푗 to 휋푖(퐺). Then {휋푖(퐺), 휙푖,푗, 퐼} is a surjective inverse system with the same inverse limit 퐺.

Example 3.2.6. A standard example of an inverse limit is given by the group of p-adic integers

+ 푖 ℤ푝. Fix a prime integer 푝. Deﬁne an inverse system {퐴푖, 휙푖,푗, 퐼} by setting 퐼 = ℤ , 퐴푖 = ℤ/푝 ℤ,

푗 푖 and 휙 : 픸푗 → 퐴푖 deﬁned by 휙 : 푎 (mod 푝) ↦→ 푎 (mod 푝) . Then

= lim /푝푖 . ℤ푝 ←− ℤ ℤ 푖∈퐼

Deﬁnition 3.2.7 ([RZ10, Ch. 2]). Suppose {퐺푖, 휙푖,푗, 퐼} is a surjective inverse system such that

퐺푖, 푖 ∈ 퐼 is a ﬁnite discrete group. Then the inverse limit

퐺 = lim 퐺 ←− 푖 푖∈퐼 is called a pro-ﬁnite group.

We have the following important fact concerning pro-ﬁnite groups.

Theorem 3.2.8 ([RZ10, Thm 2.1.3], [Ser02, I.1]). Let 퐺 be a pro-ﬁnite group. Then 퐺 is Haus- dorff, compact, and totally disconnected. Conversely, any compact, Hausdorff, totally disconnected group is pro-ﬁnite.

We will use the fact that the vertex stabilizers are pro-ﬁnite groups to show that 퐺 is locally compact.

Lemma 3.2.9. If 푋 is a locally finite tree and 퐺 = Aut(푋) is equipped with the topology defined by subbasis elements 풮(푓, 푇 ) as defined above, then 퐺 is locally compact.

Proof. It sufﬁces to show that 퐺 is locally compact at the identity. If we show that the vertex stabilizers 퐺푥 = 풮(1퐺, 푇푥), which are neighborhoods of the identity, are compact, then it follows that 퐺 is locally compact at 1퐺. 104 + Firstly, ﬁx a vertex 푥 ∈ 푉 푋. Consider the automorphism groups Aut(퐵푟(푥)), 푟 ∈ ℤ . Be-

cause 푋 is locally finite, the groups Aut(퐵푟(푥)) are finite. We may define the same topology

above Section 3.2 on Aut(퐵푟(푥)). By Lemma 3.2.3 Aut(퐵푟(푥)) is Hausdorff. The only Hausdorff

+ topology on a ﬁnite set is the discrete one, hence Aut(퐵푟(푥)) is discrete for all 푟 ∈ ℤ . Lastly, the

groups Aut(퐵푟(푥)) are all compact since they are ﬁnite.

+ Now deﬁne subgroups 퐻푟, 푟 ∈ ℤ of Aut(퐵푟(푥)) by

(︁ 푟푒푠 )︁ 퐻푟 := Im 퐺푥 −→ Aut(퐵푟(푥))

where 푟푒푠(푔) is the restriction of 푔 to the ball 퐵푟(푥). As a subspace of Aut(퐵푟(푥)), we have 퐻푟 is compact and discrete.

Let 휙푟,푠 : 퐻푠 → 퐻푟, 푟 ≤ 푠 be the continuous homomorphisms induced by the restriction map

+ of Aut(퐵푠(푥)) to Aut(퐵푟(푥)). One can easily verify then that {퐻푟, 휙푟,푠, 퐼}, 퐼 = ℤ is an inverse system. Thus we have an inverse limit

퐻 = (퐻 , 휋 ) = lim 퐻 . 푟 푟 ←− 푟 푟∈퐼

′ By Proposition 3.2.5, we know there is a corresponding surjective inverse system {휋푟(퐻), 휙푟,푠, 퐼} with the same inverse limit 퐻. Thus we may conclude from Deﬁnition 3.2.7 and Theorem 3.2.8 that 퐻 is a pro-ﬁnite group which is Hausdorff, compact, and totally disconnected.

푟푒푠 The restriction maps 퐺푥 −→ Aut(퐵푟(푥)) induce continuous homomorphisms 휓푟 : 퐺푥 →

+ 퐻푟 for all 푟 ∈ ℤ . We clearly have that 휓푟 = 휙푟,푠 ∘ 휓푠 for all 푟 ≤ 푠. Therefore, using the universal property of inverse limits (Deﬁnition 3.2.4) we have that 퐻 and 퐺푥 must be topologically isomorphic ([RZ10, prop. 1.1.1]), and so 퐺푥 is also a pro-ﬁnite group. In particular, it is compact.

Lastly, we must show that 퐺 is totally disconnected. Let 퐺0 denote the connected component

of the identity in 퐺. If we can show that this component is a singleton, i.e. 퐺0 = {id}, then by the homogeneity of 퐺, we’ll have that the connected component of every point in 퐺 is a singleton. 105 Hence it follows that 퐺 would be totally disconnected.

The following lemma gives us a condition for 퐺0 = {id}.

Lemma 3.2.10 ([Pon39, S22.G]). Let 퐺 be a topological group (assume that 퐺 is Hausdorff). If every open neighborhood 푈 of the identity element 푒 contains some open subgroup 퐻 of 퐺, then 퐺 is totally disconnected.

Proof. Let 푈 be an arbitrary open neighborhood of the identity 푒 and let 퐻 be an open subgroup with 퐻 ⊂ 푈. Note that open subgroups in a topological group are closed. Hence, we have

퐺 = 퐻 ∪ (퐺 ∖ 퐻)

and therefore 퐻 and 퐺 ∖ 퐻 form a separation for 퐺. Since 퐺0 is connected, it must lie entirely

in one of the separation sets. Since 푒∈ / 퐺 ∖ 퐻, we must have that 퐺0 ⊂ 퐻 ⊂ 푈. But because

푈 was arbitrary, we have that 퐺0 is contained in every open neighborhood of 푒. Thus we must

have that 퐺0 = {푒} since, on the contrary, if 퐺0 contains an element different than 푒, say 푓, then the constant sequence consisting only of the identity has two distinct limit points: 푒 and 푓. This is

clearly impossible in a Hausdorff space and hence we conclude that 퐺0 = {푒}.

Lemma 3.2.11. The group 퐺 = Aut(푋) is totally disconnected.

Proof. We simply must use the previous lemma and show that every open neighborhood of the identity automorphism, id퐺, contains an open subgroup. It sufﬁces to show this for every basis element 풮(푓, 푇 ), where 푓 ∈ 퐺 and 푇 ⊂ 푋 is a ﬁnite subtree, that contains id퐺. Clearly, we have that if id퐺 ∈ 풮(푓, 푇 ), then ⋂︁ 풮(푓, 푇 ) = 풮(id퐺, 푇 ) = 퐺푥 푥∈푇 Hence, any basis element that contains the identity is itself an open subgroup. Since any open neighborhood of id퐺 necessarily contains such a basis element, we may conclude that 퐺 is totally disconnected.

As one last extra item, we mention here an application of Corollary 2.8.9 from the last chapter. 106 Application 3.2.12. Consider two automorphisms 푔 and 푕 in 퐺 = Aut(푋), with |푔| > 0. Suppose → that 푕 ∈ 풮(푔, 푇 ), where 푇 is a ﬁnite subtree of 푇푔. Then we have 푔푒 = 푕푒 for every edge 푒 in 푇 , hence by the corollary, 푕 is also hyperbolic and |푕| = |푔|. Thus 풮(푔, 푇 ) is an open set in the topology of 퐺 that contains only hyperbolic automorphisms.

3.3 Other Topologies on Aut(푋)

In the previous section, we showed that for a locally ﬁnite tree, the automorphism group 퐺 = Aut(푋) is locally compact, Hausdorff, and totally disconnected, with the topology given by subbasis elements

풮(푓, 푇 ) = {푔 ∈ 퐺 | 푓|푇 = 푔|푇 }.

Let us refer to this topology on 퐺 by 휏. In this section, we show that 휏 is equivalent to several other natural topologies one might place on 퐺.

Theorem 3.3.1. Let 푋 be a locally ﬁnite tree and consider its group of automorphisms 퐺 = Aut(푋). Then the topology 휏 on 퐺 is equivalent to the topology of pointwise convergence and the compact-open topology.

Recall that, if 푋 and 푌 are topological spaces, then the topology of point-wise convergence can be placed on the set 푌 푋 of all functions from the space 푋 to the space 푌 , and is given by the following sub-basis elements:

풫(푥, 푈) = {푓 : 푋 → 푌 | 푓(푥) ∈ 푈} where 푥 is a point in 푋 and 푈 is an open set in 푌 .

The compact-open topology is placed on the set C (푋, 푌 ) of continuous functions from 푋 to 푌 , and is deﬁned by subbasis sets

풞(퐾, 푈) = {푓 ∈ C (푋, 푌 ) | 푓(퐾) ⊂ 푈} where 퐾 is a compact set in 푋 and 푈 is an open set in 푌 . Note that 풞({푥}, 푈) = 풫(푥, 푈). 107 The compact-open topology is similar to the topology of compact convergence, also called the topology of uniform convergence on compact sets. In general, the compact-open topology is ﬁner than that of compact convergence, but when 푌 is a metric space, they in fact coincide [Mun00, Theorem 46.8].

Proof. Firstly, note that automorphisms of 푋 are isometries, and thus are necessarily continuous, so it makes sense to talk about the compact-open topology on 퐺. We denote the topology of

pointwise convergence on 퐺 by 휏푝 and the compact-open topology by 휏푐. As noted above, we have

휏푐 ⊃ 휏푝, since 푋 is a metric space, 휏푐 is equivalent to compact convergence topology [Mun00, Theorem 46.8], and the compact convergence topology is ﬁner than the topology of pointwise convergence [Mun00, Theorem 46.7].

Now we show that 휏 is equivalent to 휏푝. Note that we only need to consider subbasis elements

in 휏푝 of the form 풫(푥, 퐵푟(푥)), where 푥 is a vertex in 푋 and 퐵푟(푥) is an open ball of radius 푟 centered at 푥.

Let 푥 ∈ 푋, 푟 > 0 and consider 푓 ∈ 풫(푥, 퐵푟(푥)). We want to show that there is some ﬁnite subtree 푇 of 푋 with 풮(푓, 푇 ) contained in 풫(푥, 퐵푟(푥)). Because 푋 is locally ﬁnite, open balls are

ﬁnite, thus are ﬁnite subtrees of 푋. Then clearly we have 푓 ∈ 풮(푓, 퐵푟(푥)). If 푔 ∈ 풮(푓, 퐵푟(푥)), then 푔 restricted to 퐵푟(푥) is equal to 푓 restricted to this ball; therefore 푔(푦) = 푓(푦) for all 푦 ∈

퐵푟(푥). Hence 푔 ∈ 풫(푥, 퐵푟(푥)) and we have 풮(푓, 퐵푟(푥)) ⊂ 풫(푥, 퐵푟(푥)). This shows that 휏푝 ⊂ 휏. Now let 푓 ∈ 풮(푔, 푇 ) for some automorphism 푔 and ﬁnite subtree 푇 . Note that 풮(푔, 푇 ) =

풮(푓, 푇 ). Consider the subbasis element 풫(푥, {푓(푥)} in 휏푝, where 푥 ∈ 푇 ; note that {푓(푥)} =

퐵1/2(푥), hence is an open set in 푋. Clearly 푓 ∈ 풫(푥, {푓(푥)}). If we take any 푔 ∈ 풫(푥, {푓(푥)}), then we clearly have 푔(푥) = 푓(푥).

Set 푈 = ∩푥∈푇 0 풫(푥, {푓(푥)}). Since 푇 is ﬁnite, 푈 is open. And 푈 is non-empty because 푓 ∈ 푈. Then if 푔 ∈ 푈, we have that 푔(푥) = 푓(푥) for all vertices 푥 in 푇 . Hence 푓 ∈ 푈 ⊂ 풮(푓, 푇 ), and

therefore we may conclude that 휏 and 휏푝 are equivalent topologies.

Currently, we have that 휏푐 ⊃ 휏푝 = 휏. Now all that is left to show is that 휏 is ﬁner than 휏푐 and our theorem is proved. Let 푓 ∈ 퐺 and consider the neighborhood 풞(퐾, 푈) of 푓, where 퐾 is a 108 compact set in 푋 and 푈 is an open set in 푋. Since 퐾 is compact, we can cover it by ﬁnitely many balls: 푛 ⋃︁ 퐾 ⊂ 퐵푟푖 (푥푖). 푖=1

푛 Set 퐻 = ∩푖=1풮(푓, 퐵푟푖 (푥푖)). This is a basis element for 휏 which clearly contains 푓. Let 푔 ∈ 퐻.

For any vertex 푥 in 퐾, we have that 푥 lies in some ball 퐵푟푖 (푥푖) for some 푖. Hence, we must have

that 푔(푥) = 푓(푥) since both lie in the basis element 풮(푓, 퐵푟푖 (푥푖)), which contains 푥. Therefore we have 푔(퐾) = 푓(퐾) which lies in 푈. Therefore 푔 ∈ 풞(퐾, 푈) and we have that 퐻 ⊂ 풞(퐾, 푈).

This shows that 휏 is indeed a ﬁner topology than 휏푐. Therefore, we have that 휏, 휏푝, and 휏푐 are all equivalent topologies on 퐺.

Trivial Aut(푋)

In studying the automorphism group of a tree, one generally must insist on having some conditions on the tree to ensure that there are enough automorphisms. For our purposes, this is usually done via something like the Moufang property. However, a randomly constructed locally ﬁnite tree has no non-trivial automorphisms. The following proposition should give a rough idea of why that is the case.

Proposition 3.3.2. Suppose 푋 is a locally ﬁnite tree such that distinct vertices have distinct degree: ∀푥 ̸= 푦 ∈ 푉 푋, deg(푥) ̸= deg(푦). Then Aut(푋) = {id}.

Proof. Let 푓 ∈ Aut(푋). Consider a vertex 푥 and let 푦 = 푓(푥). By deﬁnition, 푓 must send all edges in the star of 푥 to edges in the star of 푦:

푦 = 푓(푥) = 푓(훼(푒)) = 훼(푓(푒))

where 훼(푒) = 푥, 푒 ∈ 퐸푋. Hence, we have a bijection St(푥) → St(푦) between ﬁnite sets, which is only possible when 푥 = 푦. Hence, we must have that 푓(푥) = 푥 for all vertices 푥 ∈ 푉 푋. Thus 푓 = id and because 푓 was arbitrary we conclude that Aut(푋) = {id}. 109 Topological Criterion of Aut(푋)

A natural question one might ask is: when is the group 퐺 = Aut(푋) (푋 a locally ﬁnite tree) discrete? This question is provided an algorithmic answer via an appendix in [BL01, pg. 185- 212] by Hyman Bass and Jaques Tits. The key methods come from the technique of edge-indexed graphs which we discussed in Section 2.6. A result of Dan Farley [Far00] tells us that, for 푋 locally ﬁnite, the automorphism group Aut(푋) can only be one of three types of groups.

Theorem 3.3.3. Let 푋 be a locally ﬁnite tree. Then Aut(푋) is either discrete, proﬁnite, or not the inverse limit of an inverse system of discrete groups.

This comes as a corollary to his second theorem in [Far00]. Before we state it, let us make a deﬁnition. Let 푒 be an edge in 푋, a locally ﬁnite tree, and 퐻 ≤ Aut(푋). Since 푋 is a tree, the graph 푋 ∖ {푒} is disconnected, with two connected components: call them 푋훼(푒), which contains the vertex 훼(푒), and 푋휔(푒), containing 휔(푒).

′ If 푕 ∈ 퐻푒 (the stabilizer subgroup of 푒 in 퐻), we deﬁne 푕 ∈ 퐻 by

⎧ ⎪ ⎨⎪푕(푥) if 푥 ∈ 푋훼(푒) 푕′(푥) = ⎪ ⎩⎪id if 푥 ∈ 푋휔(푒)

Thus, 푕′ equals the identity on one of the connected components of 푋 ∖ {푒}, and agrees with 푕 on the other; our choice here of agreeing on 푋훼(푒) is arbitrary and is done so for simplicity.

Deﬁnition 3.3.4. With the notation as above, we say that 퐻 is edge-independent if, whenever

′ 푕 ∈ 퐻푒, we have that 푕 ∈ 퐻. Note that Aut(푋) is always edge-independent.

Now we state the precise result from [Far00].

Theorem 3.3.5. Let 푋 be a locally ﬁnite tree and 퐺 ≤ Aut(푋) be edge-independent.

1. Suppose 퐺 contains a hyperbolic automorphism. 110

(a) If 퐺푒 is trivial for some translation edge 푒, then 퐺 is discrete.

(b) If 퐺푒 is non-trivial for some translation edge 푒, then 퐺 is not the inverse limit of an inverse system of discrete groups.

2. Suppose 퐺 contains no hyperbolic automorphism (so all elements of 퐺 are elliptic).

(a) If 퐺 has a bounded orbit, then 퐺 is pro-ﬁnite.

(b) If every (hence, some) orbit is unbounded, then 퐺 is not the inverse limit of an inverse system of discrete groups.

Since Aut(푋) is always edge-independent, it is clear that Theorem 3.3.3 follows immediately from Theorem 3.3.5. If one examines the proof of Theorem 3.3.5 in [Far00], one will see that for (1a) and (2a), the hypothesis that 퐺 ≤ Aut(푋) be edge-independent is not needed. Applying what we know from Theorem 2.10.5 for nilpotent groups, we obtain the following (most likely known) result:

Theorem 3.3.6. Let 푋 be a locally finite tree. Suppose 퐺 ≤ Aut(푋) is finitely generated nilpotent. Then 퐺 is discrete or pro-finite.

Proof. Applying Theorem 2.10.5 to 퐺, there are two cases. First, suppose 퐺 has a global ﬁxed point 푥. Then the orbit of this ﬁxed point is clearly bounded. Examining the proof of Theorem 3.3.5 in [Far00], one will see that the hypothesis for edge-independence is not needed. We have that 퐺 ≤

StabAut(푋)(푥), the latter of which we know is compact by the proof of Lemma 3.2.9. Moreover, 퐺 is Hausdorff and totally disconnected, hence 퐺 must be a pro-ﬁnite group by Theorem 3.2.8. If 퐺 does not have a ﬁxed point, then there is a straight path 푇 in 푋 stable under 퐺 on which 퐺 acts by translations. Hence, all the automorphisms of 퐺 are hyperbolic (cf. Theorem 2.8.4), and thus, for any vertex 푥 ∈ 푉 푋,

퐺 ∩ 풮(id, {푥}) = 퐺푥 = {id}.

Then is must be the case that 퐺 is discrete. 111 3.4 Lattices in Topological Groups

In this section we mention brieﬂy the deﬁnition of a lattice in a locally compact topological group. References for this section include [BL01, Ch. 1], [Loo53, Sec. 29], [Fol95, Ch. 2], and [Rag72, Ch. 1]. In any locally compact group 퐺, there exists a left (or right, if you prefer) translation invariant measure 휇, called the Haar measure. It is unique up to a multiplicative positive constant. ([Loo53, Sec. 29]) Hence if 휇 is a (left) Haar measure on 퐺, then 휇(푔푈) = 휇(푈) for all measurable subsets 푈 ⊂ 퐺; and if 휈 is another (left) Haar measure on 퐺, then 휈 = 휆휇 for some real constant 휆 > 0. If 퐺 is a discrete group, then we can choose 휇 so that 휇({푔}) = 1 for 푔 ∈ 퐺.

Example 3.4.1. The Lebesque measure on (ℝ, +) is an example of a Haar measure. Likewise, the Lebesque measure on ℝ푛 (Euclidean 푛-space) is a Haar measure. The standard Lebesque measure on the unit circle 햲1 is also a Haar measure.

Example 3.4.2. On the multiplicative group (ℝ ∖ {0}, *) one has a Haar measure deﬁned by

∫︁ 1 휇(푆) = 푑푡 푆 |푡|

for 푆 a Borel subset.

On the group 퐺 = GL2(ℝ) one has a Haar measure deﬁned by:

∫︁ 1 휇(푆) = 푛 푑푇 푆 | det(푇 )|

where 푑푇 is Lebesgue measure on the vector space of all real 푛 × 푛 matrices.

Let 퐺 be a locally compact group with left Haar measure 휇. By translation invariance, if 푈 ⊂ 퐺 is measurable, 휇(푔푈) = 휇(푈). For translation on the other side, we have

휇(푈푔) = 휇(푈)Δ(푔) 112 where Δ: 퐺 → ℝ× is the modular function or modular character of 퐺 [Fol95, 2.4]. This is easy to deﬁne: ﬁx a 푔 ∈ 퐺 and consider the map

−1 휈푔 : 푈 ↦→ 휇(푈푔 )

for 푈 measurable. One can show that this is also a measure on 퐺. It is also left translation invariant:

−1 휈푔(푕(푈)) = 휇((푕푈)푔 )

= 휇(푕(푈푔−1)) by associativity of 퐺

−1 = 휇(푈푔 ) = 휈푔(푈) 휇 is left-invariant which holds for any 푕 ∈ 퐺, 푈 ⊂ 퐺 measurable. Because Haar measure is unique up to a positive real constant, we must have that there exists a Δ(푔) > 0 such that

휈푔 = Δ(푔)휇

Hence, doing this for every 푔 ∈ 퐺 yields the modular function Δ.

Deﬁnition 3.4.3. If Δ(푔) = 1, then 퐺 is said to be unimodular.

Notice immediately that 퐺 unimodular implies that the Haar measure 휇 is both left and right invariant. If 퐺 is a locally compact group and Γ is a subgroup of 퐺, then the quotient space Γ∖퐺 is locally compact (Proposition 3.1.4). If Γ is a discrete subgroup there exists an induced measure 휇Γ∖퐺 on Γ∖퐺 ([Rag72, Ch. 1]). Moreover, if we denote the quotient map 푝 : 퐺 → Γ∖퐺, then 푝 is locally measure preserving and

휇Γ∖퐺(푉 푔) = 휇Γ∖퐺(푉 )Δ(푔) 113 for any measurable 푉 ⊂ Γ∖퐺. Notice that, if 휇Γ∖퐺(Γ∖퐺) < ∞, then

휇Γ∖퐺(푔Γ∖퐺) = 휇Γ∖퐺(Γ∖퐺)Δ(푔) 휇 (Γ∖퐺) Γ∖퐺 = Δ(푔) 휇Γ∖퐺(Γ∖퐺) 1 = Δ(푔),

since of course 푔Γ∖퐺 = Γ∖퐺. So 휇Γ∖퐺(Γ∖퐺) < ∞ forces 퐺 to be unimodular. We now deﬁne lattices in this topological group setting:

Deﬁnition 3.4.4. Let Γ ≤ 퐺 be a discrete subgroup of a locally compact group 퐺. If the induced measure 휇Γ∖퐺 is ﬁnite, then Γ is called a 퐺-lattice or lattice. If the quotient Γ∖퐺 is compact, then we call Γ a uniform 퐺-lattice.

Example 3.4.5. For 퐺 = ℝ푛, the discrete subgroup ℤ푛 is an example of a uniform lattice. The quotient space is ℝ푛/ℤ푛 = 핋푛, the 푛-dimensional torus.

Example 3.4.6. The special linear groups SL푛(ℤ) < SL푛(ℝ) are non-uniform lattices. For example, in the 푛 = 2 case, we have that SL2(ℝ) acts naturally on the symmetric space ℍ =

{푥 + 푖푦 | 푥, 푦 ∈ ℝ, 푦 > 0}. A fundamental domain for the action of SL2(ℤ) is the set 퐹 = {푧 ∈ ℍ | − 1/2 ≤ ℜ(푧) ≤ 1/2, |푧| ≥ 1 (recall Section 2.2); this is a hyperbolic triangle with area

휋/3. Hence we get SL2(ℤ) a lattice, but is non-uniform as this triangle is not compact (cf. [Lub94, Example 3.2.2]).

Example 3.4.7. It was shown by Tamagawa [Tam65] that all lattices in SL2(ℚ푝) are uniform lattices (cf. [Ser80, II.1.5, p. 84]). More generally, lattices in rank one groups over a non-archimedean ﬁeld 퐾 with 푐푕푎푟(퐾) = 0 are necessarily uniform. Hence, when looking for non-uniform lattices in a group such as SL2(퐾), one must consider the case where 푐푕푎푟(퐾) > 0.

We have seen such an example: the group Γ = SL2(픽푞[푡]) is a non-uniform lattice in 퐺 =

−1 SL2(픽푞((푡 ))) (Corollary 2.5.4). A measure 휇 for 퐺 is deﬁned in [Ser80, I.1.5, p.84] and the volume 휇(퐺/Γ) is computed in [Ser80, I.1.6, p.89] (See also [BL01, Ch. 1]). 114 3.5 Tree Lattices

Let 푋 be a locally ﬁnite tree and 퐺 = Aut(푋) equipped with the topology deﬁned in Sec- tion 3.2. Hence 퐺 is locally compact and a Haar measure can be placed on 퐺; this allows one to study lattices in 퐺.

′ As we’ve seen previously, groups such as 퐻 = SL2(ℚ푝) and 퐻 = SL2(픽푝((푡))) act on 푝 + 1- regular trees, their associated Bruhat-Tits building. Lattices in 퐻 or 퐻′ are also lattices in 퐺 (where 푋 is the Bruhat-Tits tree for 퐻 or 퐻′), but 퐺 can have lattices that are not in 퐻 or 퐻′. This is one reason for the study tree lattices, which we deﬁne and discuss in this section and those to follow. For a nice account of some of the differences between lattices in groups like 퐻 and 퐻′ and 퐺, as well as lattices in semi-simple Lie groups, see [Lub95]. Before we exactly deﬁne tree lattices, we prove a basic fact concerning discrete subgroups in

퐺. Recall (Section 3.2) for 푥 ∈ 푉 푋, the stabilizer subgroup 퐺푥 is compact and open in 퐺.

Proposition 3.5.1. Let Γ be a discrete subgroup in 퐺. Then for any vertex 푥 ∈ 푉 푋, the stabilizer subgroup Γ푥 is ﬁnite.

Proof. Since Γ is discrete, so is any vertex stabilizer subgroup Γ푥. Moreover, Γ푥 is closed Proposi- tion 3.1.7. Since Γ푥 ≤ 퐺푥 and 퐺푥 is compact, we have that Γ푥 is compact. A discrete and compact

group is necessarily ﬁnite, hence Γ푥 is ﬁnite for any vertex 푥.

Now let Γ ≤ 퐺 be a discrete subgroup and suppose Γ acts on the tree 푋 without inversion of edges. Thus, we may form the quotient graph of groups Γ∖∖푋 (cf.Section 2.4). This means ∼ ∼ Γ∖∖푋 = (Γ∖푋, 풢) is a graph of groups, where 휋1(Γ∖∖푋) = Γ and 푋 = 푋̃︀, the universal covering tree of the graph of groups. In particular, the vertex and edge groups of Γ∖∖푋 can be identified with vertex and edge stabilizers in Γ. To define a tree lattice, we use the volume, as in Definition 2.3.14:

Deﬁnition 3.5.2. We deﬁne the volume of Γ∖∖푋 to be

∑︁ 1 Vol(Γ∖∖푋) = |Γ푥| 푥∈Γ∖푉 푋 ̃︀ 115 If Vol(Γ∖∖푋) < ∞, then we say Γ is an 푋-lattice. If the quotient graph Γ∖푋 is a ﬁnite graph, then we call Γ a uniform 푋-lattice. If it is understood from the context, we will just write lattice or uniform lattice. Γ is said to be a non-uniform lattice if its volume is ﬁnite but the quotient graph is not.

Remark 3.5.3. One might question why we have the above deﬁnition of a tree lattice instead of that used in Section 3.4. The above volume equation is based off a theorem of Serre [Ser80, II.1.5,

Thm. 5] for the case of SL2(퐾) where 퐾 is a complete field with discrete valuation and with finite residue field 푘; then the associated Bruhat-Tits tree is a regular tree as we have seen (cf. Section 1.3). Serre shows that one can normalize the Haar measure on 퐺 so that, for any discrete subgroup Γ < 퐺 = SL2(퐾), the measure of 퐺/Γ is exactly the volume formula above.

Example 3.5.4. Recall that from Section 2.5 we have an action of Γ = SL2(픽푞[푡]) on the Bruhat-

−1 Tits tree 푋 associated to 퐻 = SL2(픽푞((푡 ))). The action of Γ in 푋 has fundamental domain 퐹 , an inﬁnite ray Lemma 2.5.3. By deﬁnition, 퐹 ∼= Γ∖푋. Hence, to compute Vol(Γ∖∖푋) we must sum over the vertices of 퐹 .

For each vertex 푥푛 := Λ푛, 푛 ≥ 0 in the fundamental domain, we have Γ푥푛 = Γ푛 ∩ SL2(푘[푡])

2 (where Γ푛 is deﬁned as in Section 2.5). Then |Γ0| = 푞(푞 − 1) (this is the order of SL2(픽푞)) and

푛+1 |Γ푛| = (푞 − 1)푞 . We necessarily have that 푞 ≥ 2, thus the sum

∑︁ 1 (푞 − 1)푞푛+1 푛≥1

converges. We compute the volume:

1 ∑︁ 1 Vol(Γ∖∖푋) = + 푞(푞2 − 1) (푞 − 1)푞푛+1 푛≥1 1 1 = + 푞(푞2 − 1) 푞(푞 − 1)2 2 = (푞 + 1)(푞 − 1)2 2 = (푞2 − 1)(푞 − 1) 116 −1 Therefore, Γ = SL2(픽푞[푡]) is a non-uniform lattice in SL2(픽푞((푡 ))).

Recall from Section 2.7 that, if 푥˜ ∈ 푉 푋 is a lift of a vertex 푎 ∈ Γ∖푉 푋 via the projection morphism 푝 : 푋 → Γ∖푋, then we have that

|Γ푥̃| · Vol(Γ∖∖푋) = Vol푎(퐼(Γ∖∖푋)),

where we note that, in Section 2.7, we had |풜푥| instead of |Γ푥̃|; because of our identiﬁcation, these groups are isomorphic. From this we have the following conditions on the quotient graph of groups and lattices:

Proposition 3.5.5. Let Γ be a discrete subgroup of 퐺.

1. Γ is a lattice if and only if Γ∖∖푋 is a graph of ﬁnite groups, and Vol(퐼(Γ∖∖푋)) < ∞

2. Γ is a uniform lattice if and only if Γ∖∖푋 is a ﬁnite graph of ﬁnite groups.

For 푛 ≥ 3, groups such as SL푛(ℤ) and SL푛(픽푞[푡]) and lattices in SL푛(ℚ푝) are all ﬁnitely generated. This is because they have Kazhdan’s property (T) ([BdlHV08, Thm 1.6.1]), which forces

finite generation. However, we know that SL2(픽푞[푡]) is not finitely generated (Corollary 2.5.6). In fact, any non-uniform tree lattice is not finitely generated:

Proposition 3.5.6 ([BL01, 5.16]). Let 푋 be a locally ﬁnite tree and Γ a non-uniform 푋-lattice. Then:

(a) Γ has arbitrarily large ﬁnite subgroups. So in particular, Γ is not virtually torsion-free.

(b) Γ is not ﬁnitely generated.

Proof. Since Γ is non-uniform, we have

∑︁ 1 < ∞ |Γ푥̃| 푥∈푉 (Γ∖푋) 117 and the graph Γ∖푋 is inﬁnite. Thus the above sum is inﬁnite, but converges. Hence, we have that

the stabilizers Γ푥̃ must grow arbitrarily large, but they are all finite groups because Γ is discrete. This shows the first part of (a). To show that Γ is not virtually torsion-free, we proceed by contradicting. Assuming Γ is virtually torsion-free means there exists a torsion-free subgroup 푇 < Γ with finite index say [Γ : 푇 ] = 푚 < ∞. From above, we know that Γ has finite subgroups of arbitrarily large order. In particular, we know there exists some finite subgroup 퐻 of Γ such that 푚 < |퐻| < ∞. But then we have 푚 |퐻| < = [퐻 : 퐻 ∩ 푇 ] ≤ [Γ : 푇 ] = 푚 |퐻 ∩ 푇 | |퐻 ∩ 푇 |

and hence |퐻 ∩ 푇 | > 1. Thus the subgroup 퐻 ∩ 퐹 is non-trivial and finite (because 퐻 is finite). But that means there exists a non-trivial element in 푇 of finite order, meaning 푇 has torsion, which contradicts 푇 being torsion-free. Hence, we must have that Γ is not virtually torsion-free. Lastly to show (b), assume that Γ is finitely generated. Since Γ is a non-uniform lattice, it is isomorphic to the fundamental group of an infinite graph of finite groups. But if Γ is finitely generated, then it must be isomorphic to the fundamental group of a finite subgraph of finite groups. This implies that Γ must be virtually free, by a theorem in [Bas93, Thm 8.4]. Specifically, one can say:

Theorem 3.5.7. A finitely generated group 퐺 is virtually free if and only if 퐺 can be represented as the fundamental group of a finite graph of groups, where all the vertex and edge groups are finite.

The forward direction is a pretty big result by Karass, Pietrowski, and Solitar [KPS73], while the other direction is proved by Bass [Bas93, Thm 8.4] (also this is proved in [Ser80, II.2.6]). So we have that Γ is virtually free, meaning there exists a free subgroup 퐹 < Γ with finite index [Γ : 퐹 ] = 푚 < ∞. Letting 퐻 be a finite subgroup with order greater than 푚, we can use the same computation above to deduce that 퐻 ∩ 퐹 is non-trivial and finite. But now 퐻 ∩ 퐹 is a subgroup of 퐹 , which is free, and non-trivial free groups are always infinite. Thus we have arrived 118 at a contradiction, and hence we must have instead that Γ is not finitely generated. This shows (b).

3.6 Existence of Tree Lattices

Before touching on some of the structure of tree lattices, it would be good to have an idea of when exactly lattices exist. For what follows, let 퐻 ≤ 퐺 = Aut(푋) be a subgroup that acts without inversion on a locally ﬁnite tree 푋. Let 퐴 = 퐻∖푋 and edge-indexing (퐴, 푖) = 퐼(퐻∖∖푋). We label the following conditions for (퐴, 푖):

(퐴, 푖) is unimodular (U)

(퐴, 푖) has bounded denominators (BD)

Vol((퐴, 푖)) < ∞ (FV)

퐴 is ﬁnite (F)

Recall from Theorem 2.7.1 that (BD) implies (U). If (F), then in fact (BD) ⇐⇒ (U).

Let 퐺퐻 denote the subgroup of 퐺 that preserves all 퐻-orbits:

퐺퐻 = {푔 ∈ 퐺| 푔푥 ∈ 퐻푥, 푔푒 ∈ 퐻푒, ∀푥 ∈ 푉 푋, 푒 ∈ 퐸푋}

= {푔 ∈ 퐺| 푝 ∘ 푔 = 푝}

Sometimes we will write 퐺퐻 = 퐺(퐴,푖). Note that 퐻 is the largest subgroup of 퐺 such that the quotient graph 퐺퐻 ∖푋 = 퐻∖푋 = 퐴. 퐺퐻 is called the group of deck transformations.

Theorem 3.6.1 ([BK90, 4.6]). The following are equivalent:

1. (A,i) has bounded denominators (BD);

2. There exists a discrete group Φ ≤ 퐺퐻 such that Φ∖푋 = 퐻∖푋 = 퐴. 119

If both of these conditions are satisﬁed, then Φ is a uniform 퐺퐻 -lattice. Moreover, we have:

Φ is an 푋-lattice ⇐⇒ (FV)

Φ is a uniform 푋-lattice ⇐⇒ (F)

The following existence theorem is shown in a lengthy appendix of [BL01], proven by Bass, Carbone and Rosenberg. The methods used in the proof are that of edge-indexed graphs, as we deﬁned in Section 2.6 and Theorem 2.7.1.

Theorem 3.6.2 (Lattice Existence Theorem). There exists an 푋-lattice Γ ≤ 퐺(퐴,푖) if and only if (퐴, 푖) is unimodular and Vol((퐴, 푖)) < ∞ [(U) + (FV)].

Additionally, Γ can be chosen to be a uniform 퐺(퐴,푖)-lattice.

Suppose we have conditions (U) and (FV). If we replace (FV) with (F), then we obtain an existence theorem for uniform lattices:

Theorem 3.6.3 (Uniform Existence Theorem [BK90, (4.10)]). The following are equivalent:

1. 퐺 contains a uniform 푋-lattice Γ, which is also a 퐺-lattice;

2. 퐺 contains a uniform 푋-lattice Φ such that Φ∖푋 = 퐺∖푋;

3. 퐺 is unimodular and 퐺∖푋 is ﬁnite;

4. 푋 is the universal cover of a ﬁnite connected graph.

If 푋 is a locally ﬁnite tree satisfying the above conditions, then 푋 is called a uniform tree. Notice that in the above theorems, we obtain lattices, but they are necessarily uniform. Under what conditions do we obtain a non-uniform lattice? Firstly, we say a tree 푋 is rigid if its automorphism group 퐺 is discrete. If 푋 is rigid, then all 푋-lattices are uniform. So in order to obtain non-uniform lattices, we cannot have rigidity. Let 푋 be a uniform tree and suppose 퐺 = Aut(푋) is unimodular. By [BL01, Ch. 5 and 9], there exists a 퐺-invariant, minimal subtree 푋0 of 푋. Then if 푋0 is rigid, all 푋-lattices will be 120

uniform, even though 푋 is not itself rigid. We say 푋 is virtually rigid if 푋0 is rigid. This proves to be the only obstacle to obtaining non-uniform tree lattices:

Theorem 3.6.4 (Non-Uniform Lattices on Uniform Trees [Car98]). If 푋 is uniform and not virtually rigid, then 퐺 contains a non-uniform lattice Γ, which is also a non-uniform 퐺-lattice.

3.7 Structure of Tree Lattices

Here we list some results containing the structure of tree lattices. We shall not attempt to prove anything, but merely highlight the main differences between the group structure of uniform and non-uniform lattices. Proofs and additional references can be found in [BL01]. We ﬁx the following notations: 푋 is a locally ﬁnite tree and Γ ≤ 퐺 = Aut(푋) an 푋-lattice, meaning Γ is discrete and ∑︁ 1 Vol(Γ∖∖푋) = < ∞ |Γ푥| 푥∈Γ∖푋 ̃︀

Also, we denote the following subgroups:

푍퐺(Γ) = {푔 ∈ 퐺| 훾푔 = 푔훾, ∀푔 ∈ 퐺} the centralizer of Γ

−1 푁퐺(Γ) = {푔 ∈ 퐺| 푥 훾푥 ∈ Γ, ∀푔 ∈ 퐺} the normalizer of Γ

{︀ −1 −1 −1 }︀ 퐶퐺(Γ) = 푔 ∈ 퐺| [Γ : Γ ∩ 푔Γ푔 ] < ∞ and [푔Γ푔 :Γ ∩ 푔Γ푔 ] < ∞ the commensurator of Γ

(Recall in a group 퐺, two subgroups 퐴 and 퐵 are said to be commensurable if their intersection has ﬁnite index in both 퐴 and 퐵; whence the above deﬁnition of commensurator.)

Uniform Case:

∙ The quotient graph Γ∖푋 is ﬁnite.

∙ A group 퐺 is isomorphic to a uniform tree lattice ⇐⇒ 퐺 is ﬁnitely generated and virtually free (i.e. there exists 퐻 ≤ 퐺 such that 퐻 is free and [퐺 : 퐻] < ∞).

∙ 퐶퐺(Γ) is dense in 퐺. 121

∙ 푁퐺(Γ)/Γ is ﬁnite.

∙ Vol(Γ∖푋) is a rational number.

∙ For 푋 = 푋푑, the 푑-regular tree, if 푑 ≥ 3, then there exists an inﬁnite chain of uniform 푋-lattices,

Γ1 < Γ2 < Γ3 ...

and lim푛→∞ Vol(Γ푛∖∖푋) = 0.

Non-Uniform Case:

∙ Γ∖푋 is inﬁnite.

∙ Γ is not ﬁnitely generated. In fact, Γ has arbitrarily large ﬁnite subgroups.

∙ 푁퐺(Γ)/Γ is pro finite, and can be finite or infinite.

∙ 퐶퐺(Γ) may or may not be dense in 퐺.

3.8 Nagao Lattices

In this section we deﬁne a particular type of non-uniform tree lattice. Our reference is [BL01, Ch. 10]

We start with a Nagao ray, an edge-indexed graph (퐴, 푖) with index values (푞푛)푛≥0 푛 0 푒0 1 푒1 2 푒2 3 푒푛 (퐴, 푖) = 푞0 + 1 푞1 1 푞2 1 푞3 1 푞푛 1 where

푖(푒0) = 푞0 + 1, 푖(푒0) = 푞1

푖(푒푛) = 1, 푖(푒푛) = 푞푛 for 푛 ≥ 1

We require that 푞푖 ≥ 2 for 푖 ≥ 1 and 푞0 ≥ 1.

A grouping 픸 for a Nagao ray (퐴, 푖) is 122 + Γ0 Γ1 Γ2 Γ3 Γ푛

Γ0 Γ1 Γ2 Γ푛

+ such that Γ0 ≤ Γ0 , Γ0 ≤ Γ1, and Γ푖 ≤ Γ푖+1 for 푖 ≥ 1. Additionally, we set:

+ 푞0 + 1 = [Γ0 :Γ0]

푞푖 = [Γ푖 :Γ푖−1], for 푖 ≥ 1.

For this graph of groups, the monomorphisms from edge groups to vertex groups are just inclusion maps.

Now let Γ = 휋1(픸, 0), the fundamental group of this graph of groups. Then Γ is what we think of as a Nagao lattice. From Bass-Serre theory, we have that the universal covering tree 푋 = (퐴,̃ 푖, 0) admits an action by Γ, and we have

+ Γ = Γ0 *Γ0 Γ∞

(︀⋃︀ )︀ where Γ∞ = 푖≥0 Γ푖 .

This is exactly the situation we had for SL2(픽푝[푡]) in Theorem 2.5.1 and Example 3.5.4, hence the name Nagao lattice. More examples of Nagao lattices can be found in [BL01, Ch. 10]. Here we check that such a group Γ really is a non-uniform lattice, i.e. it is a discrete subgroup in Aut(푋) with ﬁnite volume. Let 퐺 = Aut(푋). By abuse of notation, we can let Γ equal its own image under the canonical homomorphism Γ → 퐺, and thus can think of Γ as a discrete subgroup in 퐺.

Clearly we have that Γ is an 푋-lattice if and only if all the Γ푖 groups are ﬁnite. Assuming then 123 that these are ﬁnite we use the equations in Section 2.7 to obtain:

(︃ )︃ 1 ∑︁ 1 1 ∑︁ 1 1 Vol(Γ∖∖푋) = + = + |Γ+| |Γ | |Γ | 푞 . . . 푞 푞 + 1 0 푛≥1 푛 0 푛≥1 1 푛 0

But since 푞 ≥ 2 for all 푛 ≥ 1, we have 1 ≤ 1 . So the sum 푖 푞푛 2

푛 ∑︁ 1 ∑︁ (︂1)︂ ≤ 푞 . . . 푞 2 푛≥1 1 푛 푛≥1 which is just a geometric series, and is ﬁnite. Hence, Vol(Γ∖∖푋) < ∞ and Γ is an 푋-lattice. Since the ray 퐴 is a fundamental domain for Γ∖푋, Γ is a non-uniform lattice. 124

CHAPTER 4 FUNDAMENTAL DOMAINS

−1 In this Chapter we study the action of 퐺 = GL2(헄[푡, 푡 ]) on its Bruhat-Tits twin-tree T = (T +, T −). From the Bass-Serre Theory developed in Chapter 2, we know that one can discern the structure of a group when it acts on a tree. In particular, if one obtains a fundamental domain for the group action, then there is an isomorphism between the group and a quotient graph of groups, built from the vertex and edge stabilizers of the fundamental domain. Hence, we seek in this chapter to discern the structure of fundamental domains for subgroups of 퐺 acting on the two halves of T .

4.1 GL2(퐾) Action on the Bruhat-Tits Tree

Throughout this section, assume 푋 is the Bruhat-Tits tree of a vector space 푉 ∼= 퐾2, with 퐾 a ﬁeld equipped with a discrete valuation as in Section 1.3. In this section, we discuss groups acting on 푋. Our main reference is [Ser80, II.1].

Let GL(푉 ) denote the group of all 퐾-automorphisms of 푉 . Given a basis {푒1, 푒2} for 푉 ,

퐺퐿(푉 ) is written GL2(퐾). The group GL2(퐾) naturally acts on the tree 푋 since GL(푉 ) acts on

2 푉 as GL2(퐾) on 퐾 by matrix-vector multiplication on the left:

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 푎 푏 푥 푎푥 + 푏푦 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ · ⎝ ⎠ = ⎝ ⎠ . 푐 푑 푦 푐푥 + 푑푦

′ Unfortunately, the group GL2(퐾) acts with inversions on 푋. Set 퐿 and 퐿 to be 풪-lattices with 125 ⎛ ⎞ 0 1 ⎜ ⎟ bases {푒1, 푒2} and {푒1, 휋푒2}, respectively. Then the matrix 푠 = ⎝ ⎠ acts by: 휋 0

⎛ ⎞ 0 1 ⎜ ⎟ ′ 푠퐿 = ⎝ ⎠ · 퐿 = 풪푒1 ⊕ 풪푒2휋 = 퐿 휋 0 ⎛ ⎞ 0 1 ′ ⎜ ⎟ ′ 푠퐿 = ⎝ ⎠ · 퐿 = 풪(휋푒1) ⊕ 풪(휋푒2) = 휋퐿 휋 0

Since 휋퐿 ∼ 퐿, we see that the element 푠 ∈ GL2(퐾) sends the edge 푒ΛΛ′ to its inverse 푒Λ′Λ. We let GL0(푉 ) denote the subgroup in GL(푉 ) of type-preserving automorphisms, where recall type was deﬁned in Deﬁnition 1.1.16. More precisely, GL0(푉 ) is the kernel of the composition

휈 ∘ det : GL(푉 ) → ℤ → ℤ/2ℤ.

Clearly GL0(푉 ) is a ﬁnite-index subgroup of GL(푉 ), with the index being 2. From [Ser80, II.1.2, Prop 1 & Cor.] we have for vertices Λ, Λ′ ∈ 푉 푋 and 푠 ∈ GL(푉 ),

푑(Λ, Λ′) ≡ 휈(det(푠)) (mod 2).

Hence GL0(푉 ) preserves types and therefore acts without inversions on 푋. By deﬁnition of the determinant and discrete valuation, we have SL(푉 ) ⊂ GL0(푉 ). The following lemma tells us that, for elements of SL(푉 ), lattice stabilizers are equivalent to vertex stabilizers.

Lemma 4.1.1 ([Ser80, II.1.3, Lemma 1]). Let 퐿 be a lattice in 푉 and Λ = [퐿] the corresponding vertex of 푋. If 퐺 is a subgroup of SL(푉 ), then 퐺퐿 = 퐺Λ.

′ Set 퐺 = SL2(퐾) for the remainder of this section. Let Λ and Λ be adjacent vertices in 푋, and

′ let 퐿 and 퐿 lattice representatives. Then we can ﬁnd a basis {푒1, 푒2} so that 퐿 = 풪푒1 ⊕ 풪푒2 and 126 ′ 퐿 = 풪푒1 ⊕ 풪(휋푒2). Then we have that,

퐺Λ = 퐺퐿 = SL2(풪),

⎛ ⎞ ⎛ ⎞ 1 0 1 0 ⎜ ⎟ ⎜ ⎟ 퐺Λ′ = 퐺퐿′ = ⎝ ⎠ SL2(풪) ⎝ ⎠ 0 휋 0 휋−1 ⎧⎛ ⎞ ⃒ ⎛ ⎞ ⎫ ⃒ ⎨⎪ 푎 푏휋−1 ⃒ 푎 푏 ⎬⎪ ⎜ ⎟ ⃒ ⎜ ⎟ = ⎝ ⎠ ⃒ ⎝ ⎠ ∈ SL2(풪) ⎪ 푐휋 푑 ⃒ 푐 푑 ⎪ ⎩ ⃒ ⎭

We can now ﬁnally mention that the group SL2(퐾) has an amalgamated free product decomposition. In [Ser80, II.1.4, Thm. 2] it is shown that the action of SL2(퐾) on 푋 has fundamental domain a segment. Hence, by our Bass-Serre theory (cf. Theorem 2.2.3), we obtain a decomposition for SL2(퐾), originally due to Ihara:

Theorem 4.1.2 ([Ser80, II.1.4, Cor. 1]). Let 퐵 < SL2(풪) deﬁned by

⎧⎛ ⎞ ⃒ ⎫ ⃒ ⎨⎪ 푎 푏 ⃒ ⎬⎪ ⎜ ⎟ ⃒ 퐵 = ⎝ ⎠ ⃒ 푐 ≡ 0 (mod 휋) . ⎪ 푐 푑 ⃒ ⎪ ⎩ ⃒ ⎭

Then we have

SL2(퐾) = SL2(풪) *퐵 SL2(풪)

Proof. From [Ser80, II.1.4, Thm 2] , we can take as fundamental domain for 퐺 = SL2(퐾) acting

′ on its Bruhat-Tits tree a segment 푇 with vertices Λ and Λ , as above. Hence, SL2(퐾) = 퐺Λ *퐺ΛΛ′

퐺Λ′ where 퐺ΛΛ′ is the ﬁxator of the edge of 푇 . ∼ From our above computations, we see that 퐺Λ = SL2(풪) = 퐺Λ′ . Moreover, 퐺Λ ∩ 퐺Λ′ = 퐵. Hence we have the decomposition for 퐺. 127

−1 4.2 The group GL2(헄[푡, 푡 ])

푒 Fix a ﬁeld 헄 = 픽푞 for some prime power 푞 = 푝 and let 핂 = 헄(푡). Recall that we can place two

+ − different valuations, 휈+ and 휈− on 핂 and obtain Bruhat-Tits trees T and T . In Section 1.4, we showed that these form a twin tree T = (T +, T −, 훿*). Let 햠 = 헄[푡, 푡−1]. Recall that we formed the twinning for T by first fixing a free 햠-module 푀 that contains a basis for 푉 = 핂2. Hence, by definition the group GL(푀) acts on T preserving co-distance, and is transitive on the set of vertices of either tree T + and T − [RT94, Sec. 2] (since we know that 퐺퐿(푀) contains elements that invert edges).

−1 Given a basis for 푀 we will write GL(푀) = GL2(헄[푡, 푡 ]). Let {푒1, 푒2} be a basis for 푀. We

ﬁx the following lattices and corresponding vertices in the twinned trees T + and T −:

푛 퐿푛,휀 = ⟨푒1, 푒2푡 ⟩풪휀

Λ푛,휀 = [퐿푛,휀]

where 휀 ∈ {+, −} denotes which tree we are referring to. We will refer to the lattices 퐿푛,휀 as

* standard lattices. In terms of the co-distance, we have 훿 (Λ푛,+, Λ푚,−) = |푛 − 푚|.

In general, for a ﬁeld 픽, the ring 푅 = 픽[푡] has the same units as 픽. For 햠, we necessarily have

푛 푛 −푛 the units from 헄 (non-zero constants), but also all 푡 , 푛 ∈ ℤ, since 푡 · 푡 = 1헄. Hence

* 푛 햠 = {푎푡 | 푛 ∈ ℤ, 푎 ∈ 헄}.

* 휀1 By deﬁnition, 푔 ∈ GL2(햠) has det 푔 ∈ 햠 . In (??) we saw that 햠 ∩ 풪휀 = 푘[푡 ], and hence

−1 GL2(헄[푡]) = GL2(햠) ∩ GL2(풪+) and GL2(헄[푡 ]) = GL2(햠) ∩ GL2(풪−).

휀 The group GL2(햠) contains automorphisms that invert edges on either tree T . That is, the 128 action of the group on the twinning does not preserve types. As in the last section, the element

⎛ ⎞ 0 푡 ⎜ ⎟ 푠 = ⎝ ⎠ 1 0

in GL2(햠) simultaneously inverts both the edges given by {Λ0,휀, Λ1,휀}, 휀 = +, −.

0 −1 We will denote the subgroup of type-preserving automorphisms by GL2([헄[푡, 푡 ]). This is deﬁned by:

0 −1 GL2(헄[푡, 푡 ]) = {푔 ∈ GL2(햠) | 휈휀(det(푔)) ≡ 0 (mod 2)} .

Note that

0 SL2(햠) ≤ GL2(햠) ≤ GL2(햠).

Stabilizers of (pairs of) lattices

1 ∼ Recall how GL2(핂) acts on the projective line ℙ (핂) = 핂 ∪ {∞} (i.e., one-dimensional subspaces of 핂2): ⎛ ⎞ 푎 푏 푎푧 + 푏 ⎜ ⎟ ⎝ ⎠ : 푧 ↦→ 푐 푑 푐푧 + 푑

1 where 푧 = [푥 : 푦] ∈ ℙ (핂). Now consider the center 푍(GL2(핂)), which is the subgroup of scalar

matrices in GL2(핂). If 푔 is a central element, then 푔 is of the form

⎛ ⎞ 휆 0 ⎜ ⎟ 푔 = ⎝ ⎠ 0 휆 with 휆 ̸= 0 ∈ 핂. Then the action of the center on ℙ1(핂) is clearly trivial:

휆푧 + 0 푔 · 푧 = = 푧. 0 · 푧 + 휆

Lemma 4.2.1 (Stabilizer of a vertex). Up to central elements we have the following 129 ⎛ ⎞ ⎛ ⎞ 1 0 1 0 1. Stab (Λ ) = ⎜ ⎟ GL (풪 ) ⎜ ⎟. GL2(핂) 푛,휀 ⎝ ⎠ 2 휀 ⎝ ⎠ 0 푡푛 0 푡−푛

⎧⎧⎛ ⎞⃒ ⎫ ⎪⎪ −푛 ⃒ ⎪ ⎪⎨⎪ 푎 푏푡 ⃒ ⎬⎪ ⎪ ⎜ ⎟⃒ ⎪ ⎜ ⎟⃒ 푎, 푏, 푐, 푑 ∈ 헄[푡] if 휀 = +, ⎪⎪⎝ 푛 ⎠⃒ ⎪ ⎪⎩⎪ 푐푡 푑 ⃒ ⎭⎪ ⎪ ⃒ ⎨⎪ 2. Stab퐺(Λ푛,휀) = ⎪⎧⎛ ⎞⃒ ⎫ ⎪ ⃒ ⎪⎪ 푎 푏푡−푛 ⃒ ⎪ ⎪⎨⎜ ⎟⃒ ⎬ ⎪ ⎜ ⎟⃒ 푎, 푏, 푐, 푑 ∈ 헄[푡−1] if 휀 = −. ⎪ ⎝ ⎠⃒ ⎪⎪ 푐푡푛 푑 ⃒ ⎪ ⎩⎩ ⃒ ⎭

+ In particular, we see that the subgroup GL2(헄[푡]) < 퐺 stabilizes the vertex Λ0,+ in T and

−1 − GL2(헄[푡 ]) < 퐺 stabilizes the vertex Λ0,− in T .

Proof. The base vertex Λ0,휀 = [⟨푒1, 푒2⟩풪휀 ] contains the standard lattice 퐿0,휀 = 풪휀푒1 ⊕풪휀푒2, which is clearly ﬁxed by GL2(풪휀). Then for Λ푛,휀 = [퐿푛,휀],

⎛ ⎞ 1 0 ⎜ ⎟ ·⟨푒 , 푒 푡푛⟩ = ⟨푒 , 푒 ⟩ ⎝ ⎠ 1 2 풪휀 1 2 풪휀 0 푡−푛 and ⎛ ⎞ 1 0 ⎜ ⎟ ·⟨푒 , 푒 ⟩ = ⟨푒 , 푒 푡푛⟩ . ⎝ ⎠ 1 2 풪휀 1 2 풪휀 0 푡푛

⎛ ⎞ ⎛ ⎞ 1 0 1 0 ⎜ ⎟ ⎜ ⎟ Hence, any element of ⎝ ⎠ GL2(풪휀) ⎝ ⎠ sends Λ푛,휀 to Λ0,휀 and then back. Since 0 푡푛 0 푡−푛

StabGL ( )(Λ0,휀) = GL2(풪휀), this shows (1). ⎛2 핂 ⎞ 푎 푏 ⎜ ⎟ If ⎝ ⎠ ∈ GL2(풪휀), then 푐 푑

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 푎 푏 1 0 푎 푏푡−푛 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ = ⎝ ⎠ . 0 푡푛 푐 푑 0 푡−푛 푐푡푛 푑 130

To get a stabilizer for 퐺 = GL2(햠), we intersect:

Stab퐺(Λ푛,휀) = StabGL2(핂)(Λ푛,휀) ∩ GL2(햠).

휀1 Since 풪휀 ∩ 햠 = 헄[푡 ], we get (2).

휀1 Note that the subgroups GL2(헄[푡 ]) of GL2(햠) are type-preserving. If we consider the stabilizers of a pair of lattices, then the above lemma immediately implies the following:

Lemma 4.2.2 (Stabilizer of a pair of lattices). Up to central elements we have the following ⎧⎛ ⎞ ⎪ ⎪ 풪+ 풪+ ⎪⎜ ⎟ ⎪⎜ ⎟ if 휀 = +, ⎪⎝ 푛 ⎠ ⎪ 푡 풪+ 풪+ ⎪ ⎨⎪ 1. StabGL2(핂)(Λ0,휀, Λ푛,휀) = ⎪ ⎪⎛ ⎞ ⎪ −푛 ⎪ 풪− 푡 풪− ⎪⎜ ⎟ ⎪⎜ ⎟ if 휀 = −. ⎪⎝ ⎠ ⎩⎪ 풪− 풪− ⎧⎧⎛ ⎞⃒ ⎫ ⎪⎪ ⃒ ⎪ ⎪⎨⎪ 푎 푏 ⃒ ⎬⎪ ⎪ ⎜ ⎟⃒ ⎪ ⎜ ⎟⃒ 푎, 푏, 푐, 푑 ∈ 헄[푡] if 휀 = +, ⎪⎪⎝ 푛 ⎠⃒ ⎪ ⎪⎩⎪ 푐푡 푑 ⃒ ⎭⎪ ⎪ ⃒ ⎨⎪ 2. Stab퐺(Λ0,휀, Λ푛,휀) = ⎪⎧⎛ ⎞⃒ ⎫ ⎪ ⃒ ⎪⎪ 푎 푡−푛푏 ⃒ ⎪ ⎪⎨⎜ ⎟⃒ ⎬ ⎪ ⎜ ⎟⃒ 푎, 푏, 푐, 푑 ∈ 헄[푡−1] if 휀 = −. ⎪ ⎝ ⎠⃒ ⎪⎪ 푐 푑 ⃒ ⎪ ⎩⎩ ⃒ ⎭

Proof. The stabilizer StabGL2(핂)(Λ0,휀, Λ푛,휀) is the intersection of StabGL2(핂)(Λ0,휀) and StabGL2(핂)(Λ푛,휀).

Similar for Stab퐺(Λ0,휀, Λ푛,휀). Using Lemma 4.2.1, the results follow.

Balls and spheres

Recall in Deﬁnition 1.3.16 we deﬁned the notion of spheres and balls in a tree. We will denote

+ 푛 − 푛 spheres around a vertex 푣 in the tree T by 햲+(푣) and for the tree T , we write 햲−(푣). Similarly for balls. 131 1 푛휀 Secondly, recall that in Proposition 1.3.20 we showed that the projective line ℙ (풪휀/풪휀푡 ),

푛 휀 푛 ≥ 1 corresponds bijectively with the sphere 햲휀 (푣), for any vertex 푣 in T . As an example, take

′ + ′ ′ 퐿0,+. If Λ is distance 푛 from Λ0,+ in T , then for a (unique) 퐿 ∈ Λ we have

′ 푛 푛 퐿 = 풪+(푥푒1 + 푦푒2) + 푡 풪+푒1 + 푡 풪+푒2

1 푛 for some point [푥: 푦] in ℙ (풪+/풪+푡 ).

휀푛 It will be convenient to introduce some notation for the many “modulo 풪휀푡 ” that we shall encounter.

Deﬁnition 4.2.3. Without loss of generality, let 휀 = +. Let 푘 ∈ ℤ≥1. We ﬁx the following notations:

헄[푡](푘) = 헄[푡]/헄[푡]푡푘

(푘) 푘 풪+ = 풪+/풪+푡

(푘) (푘) GL2 (헄[푡]) = GL2(헄[푡] )

(푘) (푘) GL2 (풪+) = GL2(풪+ ).

∑︀∞ 푖 <푘 ∑︀푘−1 푖 For any polynomial 푓 = 푓(푡) = 푓푖푡 ∈ 헄[푡], we let 푓 = 푓푖푡 . Clearly any element 푖=0 ⎛푖=0 ⎞ 푎 푏 (푘) <푘 ⎜ ⎟ of 헄[푡] is represented by a unique polynomial 푓 . Given 퐴 = ⎝ ⎠ ∈ GL(헄[푡]), we let 푐 푑 ⎛ ⎞ 푎<푘 푏<푘 퐴<푘 = ⎜ ⎟ GL(푘)(헄[푡]) 퐴<푘 ⎝ ⎠. Clearly any element of 2 is represented by a unique . We 푐<푘 푑<푘

1 (푛) (푛) shall denote the image of the pair [푥: 푦] in ℙ (풪+ ) by [푥: 푦] . And generally, we will use the notation 푋(푘) to indicate the object 푋 modulo 푡푘, wherever it makes sense. Lastly we note that we have all the same notations for + replaced with −, but with 푡 changed to 푡−1.

1 (푛) Lemma 4.2.4. Each element of ℙ (풪+ ) is represented by a unique pair in one of the following 132 forms: ⎧ ⎪ 푛−1 ⎨⎪[푥 = 1: 푦(푡) = 푦0 + ··· + 푦푛−1푡 ] or [푥, 푦] = ⎪ 푛−1 ⎩⎪[푥 = 푥1푡 + ··· 푥푛−1푡 : 푦(푡) = 1]

푛 푛−1 where 푦0, . . . , 푦푛−1, 푥1, . . . , 푥푛−1 ∈ 헄. In particular, if 헄 = 픽푞, then 햲 (푣) has (푞 + 1)푞 vertices.

푛−1 Proof. Note that 푥 = 푥1푡 + ··· + 푥푛−1푡 has valuation greater than or equal to one, and thus is not a unit.

1 (푛) (푛) Recall that the equivalence in ℙ (풪+ ) is given by multiplying by a unit in 풪 , which is any element of the form 푥 + 푟푡푛, where 휈(푥) = 0. Clearly all pairs listed above are pairwise non-equivalent.

(푛) (푛) (푛) We now have to show that any pair [푥, 푦] = [푥 : 푦 ] with 푥, 푦 ∈ 풪+ is equivalent to one

given above. Since 헄(푡) is the ﬁeld of fractions for 풪+, we can represent

푥′ 푥 = (푥′, 푥′′ ∈ 헄[푡], and 푣 (푥′) ≥ 푣 (푥′′) = 0) 푥′′ + + 푦′ 푦 = (푦′, 푦′′ ∈ 헄[푡], and 푣 (푦′) ≥ 푣 (푦′′) = 0). 푦′′ + +

′′ ′′ ′ ′′ ′ Note: we must have 휈(푥 ) = 0 as 푥 must be a unit; since 휈+(푥) = 휈+(푥 ) − 휈+(푥 ) = 휈(푥 ), we

′ ′′ ′′ * get 휈(푥 ) ≥ 0. Multiplying by the unit 푥 푦 ∈ 풪+ we see that

푥 · (푥′′푦′′) = 푥′푦′′ ∈ 헄[푡],

푦 · (푥′′푦′′) = 푦′푥′′ ∈ 헄[푡].

From our equivalence, we see that (푥(푛), 푦(푛)) ∼ (푥′푦′′(푛), 푦′푥′′(푛)) since they differ by a unit. That is, [푥 : 푦] = [푥′푦′′ : 푦′푥′′] = [푥′ : 푦′]. Thus without loss we may assume that 푥, 푦 ∈ 헄[푡], with one

(푛) of them a unit, say 푣+(푥) = 0. Then in 풪+ , the inverse of 푥 is also represented by a polynomial. Hence, (푥(푛), 푦(푛)) ∼ (1, (푦푥−1)(푛)). This means that we can replace 푥 by 1 while keeping 푦

푛 푛 some polynomial. Finally, note that 푡 헄[푡] ⊆ 푡 풪+ so that we may assume that deg(푥), deg(푦) ≤ 푛 − 1. 133 We now deﬁne the barycenter between triples of vertices.

휀 Deﬁnition 4.2.5. For any triple of vertices 푣푖 (푖 = 0, 1, 2) of T such that no vertex is on the geodesic between the remaining two, let the barycenter, denoted 푏 = bary(푣0, 푣1, 푣2), be the unique vertex lying on all geodesics between pairs of vertices from {푣0, 푣1, 푣2}. That is, {푏} = [푥, 푦] ∩ [푦, 푧] ∩ [푥, 푧]

Remark 4.2.6. We can easily show that a barycenter exists by means of induction; or, one can show its existence and uniqueness using some CAT(0) geometry. In particular, the ‘NPC’ condition ﬁrst mentioned by Bruhat and Tits (cf. [AB08, Ch. 11, Sec. 3]).

′ 휀 ′ ′ Deﬁnition 4.2.7. Let 푣, 푣 be vertices of T . Then the projection of 푣 with respect to 푣 is proj푣(푣 ),

′ ′ the unique edge on 푣 on a geodesic from 푣 to 푣 . Hence, if [푣, 푣 ] = (푒1, 푒2, . . . , 푒푛) denotes the

′ ′ 휀 geodesic from 푣 to 푣 , then proj푣(푣 ) = 푒1. So proj푣 is a function sending vertices of T to edges in StT 휀 (푥).

휀 If 퐴 ⊂ V T , we deﬁne proj퐴(푦) to be the unique vertex in 퐴 that is nearest to 푦. In particular,

′ 푛 ′ ′ for any 푛 ∈ , we have proj 푛 (푣 ) is the vertex on 햡 (푣) nearest to 푣 . Note that this is 푣 ℤ≥1 햡휀 (푣) 휀 ′ itself if 푑휀(푣, 푣 ) ≤ 푛.

푛 (푛) Example 4.2.8. If 푣 = Λ and 푤 ∈ 햲 (푣) (푛 ≥ 1) is given by [푥: 푦] , then proj 푘 (푤) is 0,+ + 햡휀 (푣) given by [푥: 푦](푘) for all 1 ≤ 푘 ≤ 푛.

This next lemma shows us the mostly obvious fact that projections must lie on geodesics.

′ 휀 ′ Lemma 4.2.9. Let 푣, 푣 be vertices in T with 푛 = 푑휀(푣, 푣 ). For each integer 푘 = 0, 1, 2, . . . , 푛,

′ ′ ′ let 푤 = proj 푘 (푣 ). Then each of the vertices 푤 lies on the geodesic [푣, 푣 ] between 푣 and 푣 . 푘 퐵휀 (푣) 푘

′ ′ Proof. Consider any such 푤푘. Assume 푤푘 does not lie on [푣, 푣 ]. Then we can let 푥 = bary(푣, 푤푘, 푣 ).

Since 푥 lies on [푣, 푤푘] we have

푑휀(푣, 푥) + 푑휀(푥, 푤푘) = 푑휀(푣, 푤푘) ≤ 푘, 134 푘 ′ and hence 푑휀(푣, 푥) ≤ 푘. So 푥 ∈ 퐵휀 (푣). Likewise, 푥 lies on [푣 , 푤푘] and thus

′ ′ 푑휀(푣 , 푥) + 푑휀(푥, 푤푘) = 푑휀(푣 , 푤푘).

′ ′ ′ Hence we have 푑휀(푣 , 푥) < 푑휀(푣 , 푤푘), contradicting the fact that 푤푘 is the nearest point to 푣 in 퐵푘(푣).

푛 Lemma 4.2.10. Suppose 푣 = Λ0,+, 푛 ∈ ℤ≥1, and 푣0, 푣1, 푣2 ∈ 햲 (푣), and let 푏 = bary(푣0, 푣1, 푣2). Then, the following are equivalent:

1. 푣 = 푏

푘 푘 푘 푘 2. 푏 = bary(푤0 , 푤1 , 푤2 ) for all 0 ≤ 푘 ≤ 푛, where 푤푗 = proj햡푘(푣)(푣푗), 푗 = 0, 1, 2.

1 3. The vertices proj햡1(푣)(푣푖) (푖 = 0, 1, 2) correspond to distinct projective points of ℙ (헄) under the correspondence with 햲1(푣). ⎛ ⎞ 푥푖 푥푗 푖 푖 (푛) 0 0 4. If 푣푖 is given by [푥 : 푦 ] , for 푖 = 0, 1, 2, then the matrices 퐴푖,푗 = ⎜ ⎟ are invertible ⎝ 푖 푗 ⎠ 푦0 푦0 푖 푖 over 풪+/푡풪+ for any 푖, 푗 ∈ {0, 1, 2} distinct. (Here we write the polynomial 푥 = 푥 (푡) =

∑︀푚 푖 푙=0 푥푙 for some 푚).

푘 Proof. If 푘 = 0, then 퐵 (푣) = {푣}, just the vertex 푣. Then proj퐵푘(푣)(푣푖) = 푣 for 푖 = 0, 1, 2 and so bary(푣, 푣, 푣) = 푣. Hence, we clearly have that (2) implies (1). To show (1) implies (2), we proceed by induction on 푘. The base case 푘 = 0 can be see above, since we clearly have 푏 = 푣 = bary(푣, 푣, 푣).

푘 푘 푘 Now suppose we know that, for some 0 ≤ 푘 ≤ 푛 − 1, we have that 푏 = bary(푤0 , 푤1 , 푤2 )

푘 푘+1 푘 푘+1 where 푤푗 = proj햡 (푣)(푣푗) for 푗 = 0, 1, 2. Let 푤푗 = proj퐵 (푣푗 ), 푗 = 0, 1, 2. Since projections 푘 푘+1 must lie on geodesics, we clearly have 푑+(푤푗 , 푤푗 ) = 1. Hence, any “new edges” one obtains 푘+1 푘+1 푘+1 푘+1 when considering the geodesics between 푤0 , 푤1 , 푤2 are those connecting the 푤푖 ’s and

푘 the 푤푖 ’s. Thus,

푘+1 푘+1 푘+1 푘 푘 푘 bary(푤0 , 푤1 , 푤2 ) = bary(푤0 , 푤1 , 푤2 ) = 푏 135 by our inductive hypothesis. So (1) implies (2).

1 Now we show (3) implies (1). Assuming (3), we know that the vertices 푤푖 = 푤푖 = proj퐵1(푣)(푣푖) correspond to distinct projective points of ℙ1(헄) under the correspondence with 햲1(푣). So in partic-

ular, 푤0 ̸= 푤1 ̸= 푤2 and each lies distance 1 away from 푣. Moreover, each 푤푖 lies on the geodesic

between 푣 and 푣푖 by the previous lemma. Thus bary(푤0, 푤1, 푤2) = 푣. By the same inductive argument used to show (1) implies (2), we can see that we must have

푘 푘 푘 푣 = bary(푤0 , 푤1 , 푤2 ) for all 0 ≤ 푘 ≤ 푛. But then we must have that 푣 = 푏 and in particular both (1) and (2) hold.

Next, suppose (1) holds and assume that the vertices 푤푖, 푖 = 0, 1, 2 do not correspond to distinct

points of ℙ1(헄). Then at least two of these vertices must be equal. Suppose that all three are the

same: 푤0 = 푤1 = 푤2. Then this vertex lies on each geodesic [푣, 푣푖] for 푖 = 0, 1, 2, and hence

must also lie on the intersection of all geodesics between any two pairs of {푣0, 푣1, 푣2}. By the

uniqueness of the barycenter, we must have 푣 = 푏 = 푤0 = 푤1 = 푤2, a contradiction of the fact

that 푤0 and 푣 lie 1 apart.

If all three 푤푖 are not equal, then by our assumption, two of them are equal. WLOG, let us say

푤0 = 푤1 but 푤2 is different. Then 푤0 lies on the intersection of geodesics [푣, 푣0] ∩ [푣, 푣1]. By (1),

we have that 푣 lies on [푣0, 푣1] and therefore so must 푤0. Likewise, 푤0 must lie on [푣0, 푣2] ∩ [푣1, 푣2].

But then this implies that 푤0 = 푏 = 푣, a contradiction. Hence, we must have that all three vertices

푤0, 푤1, 푤2 are distinct which shows (3).

푖 푖 (푛) 1 (푛) Now suppose (3) holds. Assume that 푣푖, 푖 = 0, 1, 2, is given by the point [푥 : 푦 ] in ℙ (풪 ).

1 푖 푖 (1) 푖 푖 1 Then we know the vertices 푤푖 = 푤푖 correspond to points [푥 : 푦 ] = [푥0 : 푦0] in ℙ (헄). Hence,

푖 푖 푖 푖 푖 푖 for each pair (푥0, 푦0), one of 푥0 or 푦0 is a unit in 헄. Moreover, the points 푥0 and 푦0 are linearly

independent. Hence, for any 퐴푖,푗 with 푖, 푗 ∈ {0, 1, 2} distinct, we must have

푖 푗 푗 푖 * det(퐴푖,푗) = 푥0푦0 − 푥0푦0 ∈ 헄 .

Thus the matrices 퐴푖,푗 are invertible, proving (4) follows from (3). 136

Conversely, if (4) holds, then the points 푤푖, 푖 = 0, 1, 2 must be distinct projective points. In- deed, if any two of them are the same, say 푤푖 = 푤푗 for 푖, 푗 ∈ {0, 1, 2}, then the matrix 퐴푖,푗 will be singular, contradicting (4). Hence (4) holds if and only if (3) holds. This completes the proof.

Stabilizers of spheres

푛 푛 We now study the action of the stabilizer of a vertex 푣 on 햡휀 (푣) and 햲휀 (푣), for 푛 ∈ ℤ≥0. For

0 0 the remainder of this section, ﬁx 퐺 = GL2(햠) and 퐺 = GL2(햠). Since 퐺 is transitive on vertices

휀 of T for each 휀 (although not simultaneously), we may assume that 휀 = + and that 푣 = Λ0,+. Recall (Lemma 4.2.1) that

* Stab퐺(푣) = Stab퐺0 (푣) = 햠 · GL2(헄[푡]).

⎛ ⎞ 푎 푏 ⎜ ⎟ For 퐴 = ⎝ ⎠ ∈ GL2(헄[푡]) and 푥, 푦 ∈ 헄[푡], the 풪+-module 푐 푑

푛 푛 퐴 ·⟨푥푒1 + 푦푒2 + 푡 풪+푒1 + 푡 풪+푒2⟩ is the set of vectors of the form

⎛ ⎞ ⎛ ⎞ 푎 푏 푥 + 푡푛풪 푛 푛 + 퐴 · (푥푒1 + 푦푒2 + 푡 풪+푒1 + 푡 풪+푒2) ↔ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ 푛 ⎠ 푐 푑 푦 + 푡 풪+ ⎛ ⎞ 푛 푎푥 + 푏푦 + 푡 풪+ ↔ ⎜ ⎟ ⎝ 푛 ⎠ 푐푥 + 푑푦 + 푡 풪+ ⎛ ⎞ ⎛ ⎞ <푛 <푛 <푛 <푛 푛 푎 푥 + 푏 푦 푡 풪+ ↔ ⎜ ⎟ + ⎜ ⎟ . (4.2.1) ⎝ <푛 <푛 <푛 <푛⎠ ⎝ 푛 ⎠ 푐 푥 + 푑 푦 푡 풪+

푛 (푛) 1 (푛) In other words the action of 퐺푣 on 햲 (푣) is represented by the action of 퐺푣 on ℙ (풪+ ):

(퐴[푥: 푦])(푛) = 퐴(푛)[푥(푛) : 푦(푛)] 137 푛 Lemma 4.2.11. The kernel of the action of GL2(헄[푡]) on 햲 (푣) is

⎧⎛ ⎞⃒ ⎫ ⃒ ⎨⎪ 푎 + 푡푛푎′ 푡푛푏′ ⃒ ⎬⎪ 푛 ⎜ ⎟⃒ * ′ ′ ′ ′ 푍(GL2(헄[푡])) + 푡 GL2(헄[푡]) = ⎝ ⎠⃒ 푎 ∈ 헄 , 푎 , 푏 , 푐 , 푑 ∈ 헄[푡] (4.2.2) ⎪ 푡푛푐′ 푎 + 푡푛푑′ ⃒ ⎪ ⎩ ⃒ ⎭

푛 (푛) Proof. By the remarks above, we know any 퐴 ∈ GL2(헄[푡]) acting on 햲 (푣) corresponds to 퐴

1 (푛) 푛 acting on ℙ (풪+ ). So modulo (푡 ), any matrix of the form

⎛ ⎞ 푎 + 푡푛푎′ 푡푛푏′ ⎜ ⎟ ⎝ ⎠ 푡푛푐′ 푎 + 푡푛푑′ reduces to a scalar matrix: ⎛ ⎞ 푎 0 ⎜ ⎟ * ⎝ ⎠ (푎 ∈ 헄 ), 0 푎

1 (푛) which clearly acts trivially on ℙ (풪+ ). Conversely, suppose that ⎛ ⎞ 푎 푏 ⎜ ⎟ 퐴 = ⎝ ⎠ ∈ GL(헄[푡]). 푐 푑

Let [푥 : 푦](푛) be a point in ℙ1(풪(푛)) and consider the vertex given by:

푛 푛 Λ = [⟨푥푒1 + 푦푒2 + 푡 풪+푒1 + 푡 풪+푒2⟩풪+ ].

The action of 퐴 on Λ yields vectors of the form:

⎛ ⎞ ⎛ ⎞ < < < < 푛 푎 푥 + 푏 푦 푡 풪+ ⎜ ⎟ + ⎜ ⎟ . ⎝ < < < <⎠ ⎝ 푛 ⎠ 푐 푥 + 푑 푦 푡 풪+

푛 Now suppose 퐴 is in the kernel of the action of GL2(헄[푡]) on 햲 (푣). Let 푥 = 0, 푦 = 1 above. 138 Then the action of 퐴 yields

⎛ ⎞ ⎛ ⎞ < 푛 푏 푡 풪+ ⎜ ⎟ + ⎜ ⎟ ⎝ <⎠ ⎝ 푛 ⎠ 푑 푡 풪+

which must equal

⎛ ⎞ 푡푛풪 ⎜ + ⎟ ⎝ 푛 ⎠ 1 + 푡 풪+

because 퐴 acts trivially. This forces 푏< = 0. If we let 푥 = 1, 푦 = 0, then we get that

⎛ ⎞ ⎛ ⎞ < 푛 푛 푎 + 푡 풪+ 1 + 푡 풪+ ⎜ ⎟ ≡ ⎜ ⎟ ⎝ < 푛 ⎠ ⎝ 푛 ⎠ 푐 + 푡 풪+ 푡 풪+

and thus 푐< = 0. Lastly, if we let 푥 = 1, 푦 = 1, then

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ < < 푛 < 푛 푛 푎 + 푏 + 푡 풪+ 푎 + 푡 풪+ 1 + 푡 풪+ ⎜ ⎟ = ⎜ ⎟ ≡ ⎜ ⎟ . ⎝ < < 푛 ⎠ ⎝ < 푛 ⎠ ⎝ 푛 ⎠ 푐 + 푑 + 푡 풪+ 푑 + 푡 풪+ 1 + 푡 풪+

Thus we must have 푎< = 푑<.

The following lemma is pivotal to the results that follow. It shows that the stabilizer of a barycenter has a transitive action. Preciesly:

(푛) Lemma 4.2.12. Suppose 푏 = Λ0,+, 푛 ∈ ℤ≥1. Then 퐺푏 is sharply transitive on triples (푢, 푣, 푤) of vertices in 햲푛(푏) such that 푏 = bary(푢, 푣, 푤).

Proof. Note that once transitivity is proven, the sharpness follows from the proof of Lemma Lemma 4.2.11.

1 푛 To prove transitivity, it sufﬁces to show that 퐺푏 can send the triple (푐, 푑, 푒) of points in ℙ (풪+/푡 풪+) given by 푐 = [1: 0], 푑 = [1: 1], and 푒 = [0: 1] to (푢, 푣, 푤). 139 To this end, let 푢 correspond to [푥(푢): 푦(푢)](푛) and likewise for 푣 and 푤. First note that it ⎛ ⎞ 푎 푏 ⎜ ⎟ (푛) sufﬁces to show that there exist 퐴 = ⎝ ⎠ ∈ GL2(헄[푡]) and units 휆, 휇, 휈 ∈ 풪+ so that, 푐 푑 푛 modulo 푡 풪+, we have

퐴(푛)푐(푛) = 휆푢(푛),

퐴(푛)푑(푛) = 휇푣(푛),

퐴(푛)푒(푛) = 휈푤(푛).

This means that we must have

⎛ ⎞ ⎛ ⎞ 푎<푛 푥(푢)<푛 ⎜ ⎟ <푛 ⎜ ⎟ 푛 ⎝ ⎠ ≡ 휆 ⎝ ⎠ (mod 푡 풪+) (4.2.3) 푐<푛 푦(푢)<푛 ⎛ ⎞ ⎛ ⎞ 푏<푛 푥(푤)<푛 ⎜ ⎟ <푛 ⎜ ⎟ 푛 ⎝ ⎠ ≡ 휈 ⎝ ⎠ (mod 푡 풪+) (4.2.4) 푑<푛 푦(푤)<푛

and ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 푥(푢)<푛 푥(푤)<푛 휆<푛 푎<푛 + 푏<푛 푥(푣)<푛 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ <푛 ⎜ ⎟ 푛 ⎝ ⎠ · ⎝ ⎠ = ⎝ ⎠ ≡ 휇 ⎝ ⎠ (mod 푡 풪+) (4.2.5) 푦(푢)<푛 푦(푤)<푛 휈<푛 푐<푛 + 푑<푛 푦(푣)<푛

By Lemma Lemma 4.2.10 we know that the matrix on the left is invertible modulo 푡풪+ for 푛 = 1, so that the determinant is a polynomial with non-zero constant term. That itself shows that the

푛 matrix is invertible modulo 푡 풪+ as well. More precisely, modulo 푡 we can solve Equation (4.2.5)

* using 휆0, 휈0, 휇0 ∈ 헄 = 풪+/푡풪+ (all non-zero). Any solution to this equation must have this property, so if 휆, 휇, 휈 exist, they are all units. Hence, once we’ve found 휆<푛, 휇<푛, 휈<푛 given 푥(푢), 푦(푢), 푥(푣), 푦(푣), 푥(푤), and 푦(푤), we can compute 퐴(푛). To ﬁnd 휆<푛, 휇<푛, 휈<푛 we use induction on 푛. For 푛 = 1 we can already do that as noted above. Now suppose that we already have

Equation (4.2.5) and wish to solve it for 푛 + 1. Then, we ﬁrst need to ﬁnd 휆푛, 휇푛, and 휈푛 and 140 ensure that ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 푛 푥(푢) 푥(푤) 휆 푛 푥(푣) ∑︁ ⎜ 푙 푙⎟ ⎜ 푛−푙⎟ ∑︁ ⎜ 푙⎟ ⎝ ⎠ · ⎝ ⎠ = ⎝ ⎠ 휇푛−푙 푙=0 푦(푢)푙 푦(푤)푙 휈푛−푙 푙=0 푦(푣)푙

which comes down to solving

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 푥(푢) 푥(푤) 휆 푥(푣) 푛−1 푥(푣) 푛−1 푥(푢) 푥(푤) 휆 ⎜ 0 0⎟ ⎜ 푛⎟ ⎜ 0⎟ ∑︁ ⎜ 푙⎟ ∑︁ ⎜ 푙 푙⎟ ⎜ 푛−푙⎟ ⎝ ⎠·⎝ ⎠−⎝ ⎠ 휇푛 = ⎝ ⎠ 휇푛−푙 − ⎝ ⎠·⎝ ⎠ 푦(푢)0 푦(푤)0 휈푛 푦(푣)0 푙=0 푦(푣)푙 푙=0 푦(푢)푙 푦(푤)푙 휈푛−푙

and this is easy since the ﬁrst matrix is invertible.

−1 Subtrees stabilized by subgroups of GL2(헄[푡, 푡 ])

휀 Deﬁnition 4.2.13. Fix 휀 = +, −, a subtree 푇 ⊆ T and 푠 ∈ 푇 . Deﬁne rad푇 (푠) = 푘, where 푘 is

푘 the largest integer such that 푠 = bary{푢, 푣, 푤} for some 푢, 푣, 푤 ∈ 햲휀 (푠) ∩ 푇 .

휀 For any 퐻 ≤ Aut(T ), we shall denote 퐻푇 = Stab퐻 (푇 ) the pointwise stabilizer of 푇 in 퐻.

FixT 휀 (퐻) denotes the subtree ﬁxed by 퐻. We always have the inclusion 푇 ⊂ FixT 휀 (퐻푇 ).

휀 휀 Lemma 4.2.14. Let 퐺 ≤ Aut(T ). Let 푇 be a subtree of T of ﬁnite diameter and let 퐺푇 be its

휀 stabilizer. In addition, suppose that 푇 = FixT 휀 (퐺푇 ), the subtree of T ﬁxed by 퐺푇 . Let 푆 denote a longest geodesic in 푇 . Then,

⋃︁ 푇 = 햡rad푇 (푠)(푠). (4.2.6) 푠∈푆

Proof. Let 푆 be a longest geodesic in 푇 , with vertex set 푉 푆 = {푠0, . . . , 푠푚} and 푑휀(푠푖, 푠푖+1) = 1

for 푖 = 0, 1, . . . , 푚 − 1. For any vertex 푡 ∈ 푇 ∖ 푆, we have that

푑+(푡, 푠) ≤ min{푑+(푠0, 푠), 푑+(푠푚, 푠)},

where 푠 is the vertex on 푆 nearest to 푡. Indeed, if it were the case that 푑+(푡, 푠) > 푑+(푠푖, 푠) for some

푖, then the geodesic [푡, 푠푖] would be of greater length than 푆, a contradiction. 141

Now it follows that 푠 = bary(푠0, 푡, 푠푚). Thus, by the above deﬁnition, we must have that

rad푇 (푠) 푑(푡, 푠) ≤ rad푇 (푠), which implies that 푡 ∈ 퐵 (푠). Hence,

⋃︁ 푇 ⊆ 퐵rad푇 (푠)(푠). 푠∈푆

(rad푇 (푠)) Conversely, we know by Lemma 4.2.12 that the stabilizer subgroup 퐺푠 acts sharply tran-

rad푇 (푠) sitively on triples (푢, 푣, 푤) in 햲 (푠) with 푠 = bary(푢, 푣, 푤). By deﬁnition of rad푇 (푠), there necessarily exists at least three distinct points 푢, 푣, and 푤 in 햲rad푇 (푠) ∩ 푇 with 푠 = bary(푢, 푣, 푤).

Then 퐺푇 ﬁxes these three points, and so by the sharp transitivity (Lemma 4.2.12), we must have

rad푇 (푠) that 퐺푇 ﬁxes the entire ball 햡 (푠). Hence,

⋃︁ rad푇 (푠) 햡 (푠) ⊆ FixT 휀 (퐺푇 ) = 푇. 푠∈푆

휀 휀 Corollary 4.2.15. Let 퐺 ≤ Aut(T ) be such that 푇 = 푇 = FixT 휀 (퐺) has ﬁnite diameter. Then (4.2.6) holds.

Proof. Since 퐺 ≤ 퐺푇 , the result follows.

Some facts about groups acting on graphs

Let 퐺 ≤ Aut Γ for some graph Γ.

Deﬁnition 4.2.16. A weak fundamental domain for the action of 퐺 on Γ is a subgraph 푊 such that for all 푣 ∈ VΓ and for all 푒 ∈ EΓ we have

1 ≤ |퐺 · 푣 ∩ V 푊 | < ∞ (4.2.7)

1 ≤ |퐺 · 푒 ∩ E 푊 | < ∞. (4.2.8) we call 푊 a fundamental domain if the intersection in (4.2.7) is of size 1 for all 푣 ∈ 푉 Γ. 142 Remark 4.2.17. The above definition of a fundamental domain is in fact equivalent with our previous one, Definition 2.2.1. Indeed, suppose we have a group 퐺 < Aut(Γ) for a graph Γ and 퐹 is a fundamental domain as in Definition 2.2.1. Then the natural projection map 푝 : 퐹 → 퐺∖Γ is an

isomorphism. Let 푓1, 푓2 ∈ 풪(푣) ∩ 퐹 . Then

푝(푓1) = 풪(푓1) = 풪(푣) = 풪(푓2) = 푝(푓2);

hence 푓1 = 푓2 because 푝 is injective. Thus since 푝 is an isomorphism we have that 퐹 is a fundamental domain as in Deﬁnition 4.2.16. Now suppose 퐹 is a subgraph of Γ satisfying Deﬁnition 4.2.16. Then for every vertex 푣 ∈ 푉 Γ,

there exists a unique 푓푣 ∈ 퐹 such that 풪(푣) ∩ 퐹 = {푓푣}; i.e. we have 푔푣 = 푓푣 for at least

one 푔 ∈ 퐺. it follows that 풪(푣) = 풪(푓푣) and thus 푝 : 퐹 → 퐺∖Γ is surjective. Moreover, if

풪(푓1) = 풪(푓2) for 푓1, 푓2 ∈ 퐹 , then we must have 푓1 = 푓2. Hence 푝 must be injective.

From now on 푊 denotes a weak fundamental domain for the action of 퐺 on Γ and 퐹 denotes a fundamental domain.

Deﬁnition 4.2.18. Now let 퐻 ≤ 퐺 be a subgroup of index [퐺 : 퐻] = 푚 (possibly inﬁnite). A transversal of 퐻 in 퐺 is a set trans(퐺/퐻) ((trans(퐻∖퐺)) of unique representatives for the left (right) cosets of 퐻 in 퐺.

Lemma 4.2.19. Let 퐻 ≤ 퐺 with [퐺 : 퐻] = 푚 < ∞. Then,

⋃︁ 푊퐻 = 푡 · 푊 푡∈trans(퐻∖퐺) is a weak fundamental domain for the action of 퐻 on Γ.

Proof. Let us take the convention that an element of a graph can denote a vertex or edge, unless speciﬁcally stated. Let 푇 = trans(퐻∖퐺), the right coset transversal of 퐻 in 퐺. Let 푥 ∈ Γ. We must show that

1 ≤ |퐻 · 푥 ∩ 푊퐻 | < ∞. 143

First we show that 퐻 · 푥 ∩ 푊퐻 is non-empty. Since 푊 is a weak fundamental domain, we know that 1 ≤ |퐺 · 푥 ∩ 푊 | < ∞; denote this set by {푤1, 푤2, . . . , 푤푛}. Then for some 푔 ∈ 퐺 and

−1 −1 −1 푖 = 1, 2, . . . , 푛, 푔푥 = 푤푖. Since 푔 lies in some coset of 퐻 (it lies in 퐻푔 ), we have 푔 = 푕푡 for 푡 ∈ 푇 and some 푕 ∈ 퐻. Thus, 푔푥 = 푤푖 implies

−1 푥 = 푔 푤푖

푥 = (푕푡)푤푖

−1 퐻 · 푥 ∋ 푕 푥 = 푡푤푖 ∈ 푊퐻 .

Hence, 퐻 · 푥 ∩ 푊퐻 ̸= ∅ and so 1 ≤ |퐻 · 푥 ∩ 푊퐻 |.

To show that |퐻 · 푥 ∩ 푊퐻 | < ∞, let 푧 ∈ 퐻 · 푥 ∩ 푊퐻 . As noted before,

퐺 · 푥 ∩ 푊 = {푤1, 푤2, . . . , 푤푛}, (푛 ≥ 1), and because 퐻 is of ﬁnite index 푚 in 퐺, we have |푇 | = 푚 and

|푇 · (퐺 · 푥 ∩ 푊 )| ≤ 푛 · 푚.

Now by deﬁnition of 푊퐻 , 푧 = 푡푤 for some 푡 ∈ 푇 and 푤 ∈ 푊 . Moreover, there exists an 푕 ∈ 퐻

−1 such that 푧 = 푕푥. Thus, 푤 = (푡 푕)푥 ∈ (퐺 · 푥 ∩ 푊 ). Thus 푤 = 푤푖 for some 푖 = 1, 2, . . . , 푛.

Hence, 푧 = 푡푤 = 푡푤푖 ∈ 푇 · (퐺 · 푥 ∩ 푊 ) and

퐻 · 푥 ∩ 푊퐻 ⊆ 푇 (퐺 · 푥 ∩ 푊 ).

Therefore |퐻 · 푥 ∩ 푊퐻 | ≤ 푛 · 푚 < ∞.

The following is just a special case of Lemma 4.2.19.

Corollary 4.2.20. If 퐹 is a fundamental domain for 퐺 ↷ Γ and 퐻 ≤ 퐺 with index 푚, then

|퐻 · 푣 ∩ 퐹퐻 | ≤ 푚 for all 푣 ∈ VΓ. 144 Proof. By the above lemma, we know that

⋃︁ 퐹퐻 = 푡 · 퐹 푡∈trans(퐻∖퐺) is a weak fundamental domain for 퐻 ↷ Γ. In particular, for any 푣 ∈ VΓ, |퐺 · 푣 ∩ 퐺| = 1. So in the above proof, 푛 = 1 and hence |퐻 · 푣 ∩ 퐹퐻 | ≤ 푚.

Lemma 4.2.21. Let 퐻 ≤ 퐺 ≤ Aut(Γ). Then, given any 푓 ∈ VΓ the 퐻 orbits in 퐺 · 푓 are in

bijection with the double cosets 퐻∖퐺/퐺푓 .

Proof. The vertices in the orbit 퐺 · 푓 biject to the coset space 퐺/퐺푓 via 푔푓 ↔ 푔퐺푓 . Now let

′ ′ 푡, 푡 ∈ trans(퐺/퐺푓 ). Then, 푡푓 and 푡푓 are in the same 퐻-orbit if and only if there is 푕 ∈ 퐻 with

′ ′ 푕푡푓 = 푡 푓. This happens if and only if 푕푡퐺푓 = 푡 퐺푓 for some 푕 ∈ 퐻, and this, in turn happens if

′ and only if 퐻푡퐺푓 = 퐻푡 퐺푓 .

Remark 4.2.22. Suppose we have 퐹 a fundamental domain for 퐺 ↷ Γ. Then for all 훾 ∈ VΓ, |퐺 · 훾 ∩ 퐹 | = 1. So in particular, if 푓 ∈ 퐹 ⊂ Γ, then 퐺 · 푓 ∩ 퐹 = {푓}. If 훾∈ / 퐹 , |퐺 · 훾 ∩ 퐹 | = 1 just says that there is at least one 푔 ∈ 퐺 such that 푔훾 ∈ 퐹 , and this element of 퐹 is unique.

We know from the above lemmas that, to get a weak fundamental domain for 퐻 ↷ Γ, we take the union of the images of 퐹 by transversals for 퐻∖퐺. But this yields “too large” a subgraph to

be a fundamental domain; the intersections 퐻 · 푥 ∩ 푊퐻 are too big. But if we take transversals

of double cosets 퐻∖퐺/퐺푓 , 푓 ∈ 퐹 , then we will have the right size, by the uniqueness of these elements.

Lemma 4.2.23. Suppose we are given: a fundamental domain 퐹 for 퐺 and 퐻 ≤ 퐺; and for each 푓 ∈ 퐹 , a set of unique double coset representatives 푆푓 for 퐻∖퐺/퐺푓 . Then we obtain a

fundamental domain 퐹퐻 for 퐻 setting

V 퐹퐻 = {푡푓,푠 · 푓 | 푓 ∈ 퐹, 푡푓,푠 ∈ 푆푓 }.

Proof. We follow the proof of Lemma 4.2.19. For 푥 ∈ Γ, let 푓푥 be such that 퐺 · 푥 ∩ 퐹 = {푓푥}. 145 −1 For some 푔 ∈ 퐺, 푔푥 = 푓푥. Then 푔 = 푕푡푓푥,푖푔푓푥 for some 푕 ∈ 퐻, 푔푓푥 ∈ 퐺푓푥 . Hence,

푥 = 푕푡푓푥,푖(푔푓푥 푓푥) = (푕푡푓푥,푖)푓푥

−1 ⇒ 퐻 · 푥 ∋ 푕 푥 = 푡푓푥,푖푓푥 ∈ 퐹퐻 .

So 퐻 · 푥 ∩ 퐹퐻 ̸= ∅. So now by the previous lemma, the orbits of 푥 under 퐻 are in bijection with

the unique double coset representatives 푡푓푥,푠, 푠 ∈ 푆푓푥 . Hence 1 = |퐻 · 푥 ∩ 퐹퐻 |.

+ Action of 퐺푇 on T

In this section we consider the following setup. Let 푇 ⊆ T + be a ﬁnite subtree with 푣 =

′ Λ0,+ ∈ V 푇 . Also, let 퐻 ≤ GL2(헄[푡]) ≤ Stab퐺(푣) be such that

푇 = FixT + (퐻), (4.2.9)

퐻 = Stab퐺(푇 ). (4.2.10)

We wish to determine a fundamental domain for 퐻 acting on T +.

Example 4.2.24. Let 푇 = {푣}. Then, 퐻 = Stab퐺(푇 ) = GL2(헄[푡]) has as its fundamental domain

+ + on T half of the standard apartment Σ = {Λ푛,+ | 푛 ∈ ℤ}:

+ 푅0 = {Λ푛,+ | 푛 ∈ ℕ≥0} (4.2.11)

+ + + + Proof. If Λ is any vertex of T with 푑+(푣, Λ ) = 푛, then for any 푔 ∈ 퐻 we have 푑+(푣, 푔Λ ) = 푛. By Lemma 4.2.12, 퐻 is transitive on triples of the form (푢, 푣, 푤) in 햲푛(푣). So in particular, for

+ + + Λ ∈ T with 푑+(푣, Λ+) = 푛, there exists an 푕 ∈ 퐻 such that 푕푥 = Λ푛,+. Hence we have 푅0 is a fundamental domain.

Deﬁnition 4.2.25. Let 휋푛 = {Λ푘,+ : 0 ≤ 푘 ≤ 푛}, the standard segment of length 푛. Thus 휋푛 is the minimal subtree containing these vertices. 146 Our ﬁrst lemma describes the fundamental domain for the standard segment subtrees. Note that we require the use of Lemma 4.2.12.

Lemma 4.2.26. A fundamental domain for 퐺푇 with 푇 = 휋푛 is

⋃︁ 퐹휋푛 = 푇 ∪ {푅푘 | 0 ≤ 푘 ≤ 푛},

where 푅푘 is a ray emanating from 휋푛 starting at Λ푘,+.

+ Proof. For every vertex 푥 in T , let 푝푥 denote the vertex of 휋푛 nearest to 푥 and let 푑푥 = 푑+(푥, 푝푥).

If 푥 lies on 휋푛, then clearly 푝푥 = 푥 and 푑푥 = 0.

Now suppose 푔 ∈ 퐺푇 . Then 푑+(푥, 푔푥) = 2 · 푑+(푥, 푝푥) = 2푑푥. Hence, 푝푥 = 푝푔푥 and 푑푥 = 푑푔푥. Denote the set of vertices with the same nearest point and distance as 푥 by

+ 푍(푥) := {푧 ∈ V T | 푝푧 = 푝푥 and 푑푧 = 푑푥}.

Clearly we have 퐺푇 · 푥 ⊆ 푍(푥). We claim that the inverse inclusion holds also. If 푥 ∈ 푇 , then 푍(푥) = {푥} and the inclusion is obvious. If 푥∈ / 푇 , then if 푦 ∈ 푇 , we have

푝푦 = 푦 and 푑푦 = 0; thus 푦∈ / 푍(푥). So we may assume that 푥∈ / 푇 and if 푧 ∈ 푍(푥), then 푧∈ / 푇 also.

So let 푧 ∈ 푍(푥). Then 푝푥 = 푝푧 and 푑푥 = 푑푧. Set 푚 = max(푛, 푑푥) and consider the

푚 ′ 푚 sphere 햲 (푝푥). Choose vertices 푢, 푣, 푣 , and 푤 in 햲 (푝푥) such that 푝푢 = Λ0, 푝푤 = Λ푛, 푧 lies

′ ′ ′ on [푣 , 푝푥], and 푥 lies on [푣, 푝푥] (hence 푝푣 = 푝푣 = 푝푥 = 푝푧). Then we have 푝푥 = bary(푢, 푣 , 푤) =

′ bary(푢, 푣, 푤) = 푝푧. By Lemma 4.2.12, 퐺푝푥 can send (푢, 푣 , 푤) to (푢, 푣, 푤). Hence, there exists

′ 푔 ∈ 퐺푝푥 such that 푔푢 = 푢, 푔푣 = 푣 , and 푔푤 = 푤. Because 푢 and 푤 are ﬁxed and 푇 ⊂ [푢, 푤], 푔

′ also ﬁxes 푇 pointwise. Thus 푔 ∈ 퐺푇 . Hence, 푔 sends the geodesic [푣, 푝푥] to the geodesic [푣 , 푝푥]

and thus sends 푥 to 푔푥 = 푧. Thus 푧 ∈ 퐺푇 · 푥 and we have 퐺푇 · 푥 = 푍(푥).

So we have that any vertex 푥 can be sent to a vertex 푧 in 푅푘, 0 ≤ 푘 ≤ 푛 where 푝푥 = 푝푧 = Λ푘,+

+ and 푑푥 = 푑푧. Hence, any vertex of T can be mapped into 퐹휋푛 . The fact that vertices of 퐹휋푛 are 147 pairwise inequivalent follows the same argument as the previous example.

Proceeding as in the previous lemma, we obtain fundamental domains for balls centered at

+ Λ0,+. Recall that, because our action on T is transitive, the point Λ0,+ is used simply for convenience.

푛 Lemma 4.2.27. A fundamental domain for 퐺푇 with 푇 = 햡 (Λ0,+) is

푛 (︁⋃︁ 푛 )︁ 퐹햡푛 = 햡 (Λ0,+) ∪ {푅푠 | 푠 ∈ 햲 (Λ0,+)} ,

푛 where 푅푠 is a ray emanating from 햡 (Λ0,+), starting at 푠.

+ 푛 Proof. We proceed as in the previous proof. Let 푥 ∈ V T and 푝푥 ∈ 햲 (Λ0,+) nearest to 푥, with

푑푥 = 푑+(푥, 푝푥).

+ As above, we have that 퐺푇 · 푥 ⊆ 푍(푥). Let 푧 ∈ 푍(푥). Since our tree T is thick, we can

guarantee the existence of at least two more edges lying on Λ0,+, other than projΛ0 (푥). Hence, we

푛 can choose points 푢 and 푤 so that 푑푢 = 푑푤 = 푑푥, with 푝푢 ̸= 푝푤 ∈ 햲 (Λ0,+) (and ̸= 푝푥) and points

′ ′ ′ 푣, 푣 with 푥 ∈ [푥, 푝푥] and 푧 ∈ [푣 , 푝푥 = 푝푧]. Then Λ0,+ = bary(푢, 푣, 푤) = bary(푢, 푣 , 푤) and so by

′ ′ Lemma 4.2.12, ∃푔 ∈ 퐺Λ0,+ sending (푢, 푣, 푤) ↦→ (푢, 푣 , 푤). Moreover, the points 푝푢, 푝푣 = 푝푣 , 푝푤

푛 푛 in 햲 (Λ0,+) are necessarily ﬁxed by 푔. So by the sharply transitiveness of 퐺Λ0,+ on 햲 (Λ0,+), any

푛 ′ such 푔 ﬁxes all of 햡 (Λ0,+), yet sends 푣 to 푣 . Hence, 퐺푇 · 푥 = 푍(푥). By earlier arguments, we

+ can conclude that 퐹휋푛 is a fundamental domain for 퐺푇 ↷ T .

Our last result gives a description of fundamental domains for any subtree 푇 as described at the beginning of this section.

Lemma 4.2.28. Suppose that 햡 = 햡푚(푣) (푣 ∈ V 푇 ) is the smallest ball with center on 푇 contain-

ing 푇 . Let 퐹퐵 be a fundamental domain for 퐺푇 acting on 햡 − 푇 . Then, a fundamental domain for 148 + 퐺푇 on T is

퐹 = 푇 ∪ 퐹퐵 ∪ {푅푓 | 푓 ∈ 햲 ∩ (푇 ∪ 퐹퐵)}, (4.2.12)

푚 where 햲 = 햲 (푣) and 푅푓 is a ray emanating from 햡 and starting at 푓.

Proof. Since 퐺푇 · 햡 = 햡 we have that the quotient graph 퐺푇 ∖햡 is a tree and so a fundamental domain 퐹퐵 exists.

Let 푥 ∈ V T +. There are three cases.

1. If 푥 ∈ 푇 , then 퐺푇 · 푥 = {푥}. Thus |퐺푇 · 푥 ∩ 퐹 | = 1.

2. If 푥 ∈ 햡 ∖ 푇 , then by assumption we have a unique 푓 ∈ 퐹퐵 such that 퐺푇 · 푥 ∩ 퐹퐵 = {푓}.

Hence |퐺푇 · 푥 ∩ 퐹 | = 1.

3. Now suppose 푥∈ / 햡. Let 푝푥 ∈ 햲 be the nearest point to 푥 in 햡. Then we know that there

is a unique vertex 푓 ∈ 퐹퐵 such that 퐺푇 · 푝푥 ∩ 퐹퐵 = {푓}. So let 푔 ∈ 퐺푇 with 푔푝푥 = 푓.

′ Since 푝푥 ∈ 햲, so is 푓. By the proof of Lemma 4.2.27, there exists 푔 ∈ 퐺퐵 ≤ 퐺푇 such that

′ 푔 푔푥 ∈ 푅푓 , a ray emanating from 푓. Note that

푑+(푥, 푝푥) = 푑푥 = 푑+(푔푥, 푓)

′ = 푑+(푔 푔푥, 푓).

′ ′ Hence, the vertex 푥 = 푔 푔푥 ∈ 푅푓 distance 푑푥 from 푓 is the unique vertex on 푅푓 at this

′ distance. Hence, 퐺푇 · 푥 ∩ 퐹 = {푥 }.

Combining all three cases, we have that 퐹 is a fundamental domain. 149 4.3 The action on T −

Recall in Section 2.5 we saw that the action of GL2(헄[푡]) on a tree yielded an amalgamated free product decomposition:

GL2(푘[푡]) = GL2(푘) *퐵(푘) 퐵(푘[푡]).

At the time, we phrased this as a nice application of Bass-Serre theory and the Bruhat-Tits tree construction. But Nagao’s theorem is a much more enlightening result when seen through the lens of twin-trees.

− In Section 2.5, we had GL2(헄[푡]) act on the Bruhat-Tits tree T - the tree associated with the valuation at inﬁnity on 핂 = 헄(푡). The tree T − is part of the twin tree T = (T +, T −, 훿*) and as

+ we saw in Lemma 4.2.1, GL2(헄[푡]) is the stabilizer of a vertex in T . One can show, by the same

−1 + arguments as in [Ser80, II.1.6], that GL2(헄[푡 ] acting on the tree T has the same fundamental domain structure as in Nagao’s Theorem - i.e., an inﬁnite ray. This observation that a stabilizer of a vertex acting on the opposite tree and yielding a ray as fundamental domain can be generalized to groups acting on twin trees. Firstly, we must deﬁne the appropriate setting for a group acting on a twin-tree. Clearly a group 퐺 acting on a twin tree should

−1 act as a twin-tree automorphism (Section 1.4). For example, GL2(헄[푡, 푡 ) acts on the Bruhat-Tits

−1 tree T mentioned above. However, the action of GL2(헄[푡, 푡 ) is not type-preserving when acting on T . We prefer type-preserving actions because this implies that the action will not invert edges. Even more favorably, we would like some kind of transitivity for our action. This leads to the following deﬁnition.

Deﬁnition 4.3.1. We say a group 퐺 acts strongly transitively on a twin tree 푇 = (푇 +, 푇 −, 훿*) if the action of 퐺 is type-preserving and if the action on the set {(푐+, 푐−) ∈ 퐸푇 + × 퐸푇 − | 푐+ opp 푐−} is transitive.

Proposition 4.3.2. A group 퐺 acting strongly transitively on a twin tree 푇 = (푇 +, 푇 −, 훿*) also acts transitively on pairs of opposite vertices of the same type.

Proof. Recall in Lemma 1.4.5 we saw that every vertex has at least one opposite provided the trees 150 푇 휀, 휀 ∈ {+, −} are at least 2-regular. Hence we assume this condition. Now let (푥+, 푥−) and (푣+, 푣−) be pairs of opposite vertices of the same type. Choose any vertex 푦+ adjacent to 푥+. Then 훿*(푦+, 푥−) = 1, so by the (Codist) property, there exists a unique vertex 푧− adjacent to 푥− such that 훿*(푧−, 푦+) = 2. Hence, any other vertex 푦− adjacent to 푥− will be opposite 푦+. Choosing any such 푦− we now have opposite edges 푐+ = {푥+, 푦+} and 푐− = {푥−, 푦−}. Similarly, we can ﬁnd opposite edges 푑+ and 푑− lying on the vertices 푣+ and 푣−. Then by the strong transitivity of 퐺, we can map the pair (푐+, 푐−) to the pair (푑+, 푑−). Since 퐺 is type-preserving, we necessarily send 푥+ to 푣+ and 푥− to 푣−.

In [AB08, 6.3, Lemma 6.70], several equivalent deﬁnitions for strongly transitive actions are given. One equivalence for 퐺 acting strongly transitively is for 퐺 to act transitively on {(푥+, 푥−) | 훿*(푥+, 푥−) = 푁} for any 푁.

−1 Example 4.3.3. Although the group GL2(헄[푡, 푡 ]) does act transitively on the set of pairs of op-

posite edges in the twin tree T , it is not type-preserving; hence the action is not strongly transitive.

−1 However, the subgroup SL2(헄[푡, 푡 ]) is type-preserving, and acts strongly transitively on T (cf.

−1 [Abr96]). But, SL2(헄[푡, 푡 ]) does not satisfy the conclusion of Lemma 4.2.12. As a middle ground

0 1 of sorts, we have that GL2(헄[푡, 푡 ]) acts strongly transitively on T (cf. [RT94, Sec. 4]).

The following theorem of Abramenko generalizes the above observation.

Theorem 4.3.4 ([Abr96, p. 52]). Let 퐺 be a group acting strongly transitively on a twin tree

+ − * − − 푇 = (푇 , 푇 , 훿 ). Let 푥 be a vertex or edge in 푇 and set Γ = 퐺푥− . Choose a twin apartment Σ = (Σ+, Σ−) with 푥− ∈ Σ−, and let 푥+ denote the vertex or edge in Σ+ opposite 푥−. Denote

+ + + + + the vertices of Σ by . . . , 푥−1, 푥0 , 푥1 , 푥2 ,... (i.e. from one end of the apartment to the other). Set

+ + + + Γ푖 = Γ푥푖 , 푖 ∈ ℤ. Assuming that 푥 = 푥0 or the edge {푥−1, 푥0 }, we have the following:

− ⋃︀∞ 1. If 푥 is a vertex, then Γ = Γ0 *Γ0∩Γ1 ( 푖=1 Γ푖) with Γ푖 ⊂ Γ푖+1 for all 푖 ∈ ℕ.

− (︀⋃︀−∞ )︀ ⋃︀∞ 2. If 푥 is an edge, then Γ = 푖=−1 Γ푗 *Γ−1∩Γ0 ( 푖=0 Γ푖), with Γ푖 ⊂ Γ푖+1 for 푖 ∈ ℤ≥0 and

Γ−푖 ⊂ Γ−푖−1 for all 푖 ∈ ℕ. 151 − By examining Section 2.5 above, one can see that GL2(헄[푡]) acting on the tree T yields exactly the situation in (1) (albeit with the roles of + and − reversed). Precisely, we have 퐺 =

0 1 GL2(헄[푡, 푡 ]) acting strongly transitively on the twin tree T , with Γ = 퐺Λ0,+ . The theorem above is applied by Abramenko and Remy [AR09] to show that, for certain types of thick locally ﬁnite twin trees 푇 = (푇 +, 푇 −, 훿*), the group Γ in the theorem is a 푇 +-lattice of Nagao type as deﬁned in Section 3.8. The twin trees for which this applies are called Moufang twin trees (cf. [RT94, Sec. 4]). These are trees with big automorphism groups.

Example 4.3.5. Let T = (T +, T −, 훿*) be the twin tree associated to a free 헄[푡, 푡−1]-module 푀

− of rank 2. Let {푒1, 푒2} be a basis for 푀. Recall the standard apartment Σ = {Λ푛,− | 푛 ∈ ℤ},

where Λ푛,− = [퐿푛,−] with

푛 퐿푛,− = 풪−푒1푡 ⊕ 풪−푒2.

Nagao’s Theorem (Section 2.5) shows that the ray (Λ0,−, Λ−1,−, Λ−2,− ...) is the fundamental do-

− main for the action of GL2(헄[푡]) on T (note in Lemma 2.5.3 we labelled with vertices with positive subscripts, but these are the same lattices). Recall the vertex stabilizers were the upper triangular subgroups:

⎧⎛ ⎞⃒ ⎫ ⃒ ⎨⎪ 푎 푏 ⃒ ⎬⎪ ⎜ ⎟⃒ * Γ−푛 = ⎝ ⎠⃒ 푎, 푑 ∈ 푘 , 푏 ∈ 푘[푡], deg(푏) ≤ |푛| . ⎪ 0 푑 ⃒ ⎪ ⎩ ⃒ ⎭ with 푛 ≤ −1.

Similarly, one can show that the ray (Λ0,−, Λ1,−,...) is also a fundamental domain for the

− action of GL2(헄[푡]) on T , by the same methods as in Section 2.5. In this case, the stabilizers of the vertices Λ푛,−, 푛 ≥ 0 are given by

⎧⎛ ⎞ ⃒ ⎫ ⃒ ⎨⎪ 푎 0 ⃒ ⎬⎪ ⎜ ⎟ ⃒ * Γ푛 = ⎝ ⎠ ⃒ 푎, 푑 ∈ 헄 , deg(푐) ≤ 푛} . ⎪ 푐 푑 ⃒ ⎪ ⎩ ⃒ ⎭ 152 + 0 −1 Let 퐵 denote the subgroup of GL2(헄[푡, 푡 ]) given by

⎧⎛ ⎞ ⃒ ⎫ ⃒ ⎨⎪ 푎 푏 ⃒ ⎬⎪ + ⎜ ⎟ ⃒ 퐵 = ⎝ ⎠ ∈ GL2(헄[푡]) ⃒ 푐 ≡ 0 (mod 푡) . ⎪ 푐 푑 ⃒ ⎪ ⎩ ⃒ ⎭

+ + Note that 퐵 is the ﬁxator subgroup in GL2(헄[푡]) of the edge {Λ0,+, Λ1,+} in the positive tree T (cf. Lemma 4.2.2).

Σ+ Λ−2,+ Λ−1,+ Λ0,+ Λ1,+ Λ2,+ Λ3,+

Σ− Λ2,− Λ1,− Λ0,− Λ−1,− Λ−2,− Λ−3,−

Since the geodesic rays {Λ푗,−}푗≥0 and {Λ푗,−}푗≤0 can both serve as a fundamental domain fors

+ GL2(헄[푡]) acting on T , we have that under the action of 퐵 the vertices Λ푛,−, 푛 ≥ 0 are pairwise

+ 0 −1 inequivalent modulo 퐵 , as are the vertices Λ푛,−, 푛 ≤ 0. To see this, recall that GL2(헄[푡, 푡 ]) preserves the co-distance function 훿*. In particular we have co-distances

* 훿 (Λ푛,−, Λ0,+) = |푛|, and

* 훿 (Λ푛,−, Λ1,+) = |푛 − 1|,

+ for all 푛 ∈ ℤ. Since the subgroup 퐵 ﬁxes Λ0,+ and Λ1,+, the image of any vertex Λ푛,− under the

+ action of 퐵 must have codistance |푛| with Λ0,+ and codistance |푛 − 1| with Λ1,+. This shows the pairwise inequivalence of vertices in Σ− under the action of 퐵+.

+ If we take the subgroups Γ푛, 푛 ∈ ℤ and intersect with 퐵 , we will obtain the vertex stabilizers under the 퐵+ action. These are given by:

⎧⎛ ⎞⃒ ⎫ ⃒ ⎨⎪ 푎 푏 ⃒ ⎬⎪ ⎜ ⎟⃒ * Δ푛 = ⎝ ⎠⃒ 푎, 푑 ∈ 푘 , 푏 ∈ 푘[푡], deg(푏) ≤ |푛| ⎪ 0 푑 ⃒ ⎪ ⎩ ⃒ ⎭ 153 for 푛 ≤ −1; for 푛 = 0 we have

⎧⎛ ⎞ ⃒ ⎫ ⃒ ⎨⎪ 푎 푏 ⃒ ⎬⎪ ⎜ ⎟ ⃒ * Δ0 = 퐵(헄) = ⎝ ⎠ ⃒ 푎, 푑 ∈ 헄 , 푏 ∈ 헄 ; ⎪ 0 푑 ⃒ ⎪ ⎩ ⃒ ⎭ and for 푛 ≥ 1,

⎧⎛ ⎞⃒ ⎫ ⃒ ⎨⎪ 푎 0 ⃒ ⎬⎪ ⎜ ⎟⃒ * Δ푛 = ⎝ ⎠⃒ 푎, 푑 ∈ 헄 , 푐 ∈ 헄[푡], deg(푐) ≤ 푛, 푐 ≡ 0 (mod 푡) . ⎪ 푐 푑 ⃒ ⎪ ⎩ ⃒ ⎭

Note that Δ0 ∩ Δ1 = 퐵(헄) and Δ0 ∩ Δ1 = 퐷(헄), where 퐷(헄) denotes the subgroup of scalar matrices. Now by applying the Bass-Serre theory Section 2.4, we obtain an amalgam for 퐵+ (we run the amalgam from 푛 = ∞ to 푛 = −∞ since that is how we naturally order the apartment Σ−):

(︃ ∞ )︃ (︃ ∞ )︃ + ⋃︁ ⋃︁ 퐵 = Δ푗 *퐵(헄) Δ푗 . 푗=0 푗=−1

Hence, we have just proved a special case of Theorem 4.3.4 for the twin tree T via the action

0 −1 of GL2(헄[푡, 푡 ]).

4.4 Future Work

In this chapter we have detailed the structure of fundamental domains for certain subgroups of

−1 + 퐺 = GL2(헄[푡, 푡 ]). Speciﬁcally, we have considered the case where 푇 is a ﬁnite subtree of T

containing a vertex 푣 = Λ0,+, 퐻 ≤ GL2(헄[푡]) ≤ Stab퐺(푣), and 푇 = FixT + (퐻). In Section 4.2 we determined the structure of fundamental domains for the action of 퐻 on T +. The next step is to consider the action of 퐻 on T − and determine a fundamental domain for this action. For any vertex 푥− in T − and 푥 ∈ 푇 we have

훿*(푥, 푥−) = 훿*(푥, 푕푥−)

− for all 푕 ∈ 퐻. Hence, the co-distances between vertices in the orbit 풪퐻 (푥 ) and 푥 ∈ 푉 푇 are all 154

equal. Using our knowledge of the structure of a fundamental domain 퐹퐻 for the action of 퐻 on

T +, we would like to obtain a similar description for the action of 퐻 on T −. This will be the focus of future work. From Theorem 4.3.4, we know the structure of a fundamental domain for a general group acting strongly transitively on a twin tree and in the case when 푇 is a vertex or edge. The main goal of the

−1 above special case of 퐺 = GL2(헄[푡, 푡 ]) is to see what can be done to extend this result for other instances of a ﬁnite subtree 푇 . As the work in Section 4.2 relied heavily on transitivity properties (cf. Lemma 4.2.12), one will most likely need similar hypothesis for the general case. 155

CHAPTER 5 CHABAUTY SPACES

At the end of Chapter 4, we saw that when a group acts strongly transitively on a twin-tree, one obtains non-uniform tree lattices via vertex stabilizers subgroups (Theorem 4.3.4). Each of these vertex stabilizer subgroups is a discrete group, hence closed (Proposition 3.1.7). This leads to the idea of studying lattices in groups acting on twin trees by ﬁrst examining the topological properties of the collection of all closed subgroups. For any topological group 퐺, the collection of closed subgroups 풞(퐺) can be endowed with a natural compact topology called the Chabauty topology. This chapter begins with an introduction to the general idea of chabauty spaces, and we also discuss metrizability properties for topological groups. In particular, we construct a left-invariant metric for 퐺 = Aut(푋) where 푋 is a locally ﬁnite tree, and ‘lift’ this metric to a metric on the space of closed subgroups of 퐺. We utilize this metric to further understand the Chabauty space associated to a group acting on a twin-tree.

5.1 Chabauty Topology

Let 푋 be a topological space. Denote by ℱ(푋) the collection of closed subsets in 푋. There is a natural topology on ℱ(푋) that makes it into a compact space. It was first developed by Chabauty in [Cha50] and hence is called the Chabauty topology. Seemingly unaware of the Chabauty article, Fell in [Fel62] also defines this topology, calling it the ℋ-topology, so some older publications may refer to the Fell topology. Presently, it appears everyone has formed a consensus on Chabauty as the appropriate nomenclature. See de la Harpe’s note [dlH08] for more history on this topology. We define the Chabauty topology on ℱ(푋) via a sub-basis, formed by the following two types of sets:

풪퐾 = {퐹 ∈ ℱ(푋) | 퐹 ∩ 퐾 = ∅},

′ 풪푈 = {퐹 ∈ ℱ(푋) | 퐹 ∩ 푈 ̸= ∅}, 156 where 퐾 is a compact subset of 푋 and 푈 is a non-empty open set in 푋. The basis is formed by taking arbitrary unions of ﬁnite intersections of these sets. So a typical basis element is of the form:

′ ′ 풪퐾1 ∩ · · · ∩ 풪퐾푛 ∩ 풪푈1 ∩ · · · ∩ 풪푈푚

with 푛, 푚 ≥ 0 (where = 0 means not included). Note that 풪퐾1 ∩ · · · ∩ 풪퐾푛 = 풪퐾1∪···∪퐾푛 . Since a ﬁnite union of compact sets is again compact, any basis element can always be written in the form

′ ′ 풪퐾 ∩ 풪푈1 ∩ · · · ∩ 풪푈푚

where 퐾 is compact. We will often denote such a basis element by ℬ(퐾, 풰), where 풰 denotes a ﬁnite collection of non-empty open sets. Hence, to say a closed set 퐹 ∈ ℬ(퐾, 풰), we mean 퐹 ∩ 퐾 = ∅ and, for each 푈 ∈ 풰, 퐹 ∩ 푈 ̸= ∅.

′ It would do us good to actually show that sets of the form 풪퐾 and 풪푈 do in fact form a subbasis. All we need is that they generate the set ℱ(푋). Since the empty set is trivially a compact

set, every 퐹 ∈ ℱ(푋) is contained in 풪∅. For any (non-empty) 퐹 ∈ ℱ(푋), we have 퐹 ∩ 푋 = 퐹 ,

′ ⋃︀ ′ and so 퐹 ∈ 풪푋 . So clearly we have ℱ(푋) = 퐾,푈 풪퐾 ∪ 풪푈 when we let 퐾 and 푈 run over all compact and non-empty open sets.

Let us state our ﬁrst result concerning the Chabauty topology.

Theorem 5.1.1. Let 푋 be a topological space. The space of closed subsets ℱ(푋) is compact with the Chabauty topology.

The proof below is from [Pau07], which is in French but also a similar proof can be found in [CEG87] or the reprint [CEG06], both of which are in English.

Proof. Since we have a description of the topology on ℱ(푋) in terms of a sub-basis, it sufﬁces to show (via the Alexander Subbase Theorem) that, given a covering of the form

{풪 , 풪′ } 퐾푖 푈푗 푖∈퐼,푗∈퐽 157 of ℱ(푋), there is a ﬁnite sub-collection that also covers ℱ(푋). From the given covering, we have a collection of open sets in 푋, {푈푗}푗∈퐽 . The union of all sets 푈푗 is open in 푋, hence the complement ⋃︀ is closed. Setting 퐹 = 푋 ∖ 푗∈퐽 푈푗, we have a closed set in 푋 that is not covered by any of the 풪′ {풪 , 풪′ } ℱ(푋) 푈푗 sets. But since the collection 퐾푖 푈푗 푖∈퐼,푗∈퐽 covers , there must exist some compact set 퐾 , 푖 ∈ 퐼 such that 퐹 ∈ 풪 . Thus the sets 퐹 and 퐾 do not intersect, hence 푖0 0 퐾푖0 푖0

⋃︁ 퐾푖0 ⊂ (푋 ∖ 퐹 ) = 푈푗 푗∈퐽

So we see that the collection {푈푗}푗∈퐽 forms an open covering of the compact set 퐾푖0 , thus ﬁnitely

many of them, say 푈푗1 , . . . , 푈푗푛 will sufﬁce to cover 퐾푖0 . And now we can conclude the following:

′ ′ ℱ(푋) = 풪푘 ∪ 풪 ∪ · · · ∪ 풪 푖0 푈푗1 푈푗푛

This follows from the fact that, if 퐸 ∈ ℱ(푋) does not intersect any of the 푈푗1 , . . . , 푈푗푛 sets, then 퐸 cannot intersect 퐾 either, since these open sets cover 퐾 . So 퐸 ∈ 풪 . Thus any closed set 푖0 푖0 퐾푖0 푋 푈 ′ 푠 퐾 in will either intersect one of the 푗푘 , or not intersect 푖0 . Our arbitrary covering has yielded a ﬁnite sub-covering, therefore we have that ℱ(푋) is compact.

We see that the Chabauty topology is compact no matter the underlying topological space 푋. In general, ℱ(푋) will not be Hausdorff - even if the underlying space is metrizable. However, when 푋 is locally compact, we obtain Hausdorff separation for ℱ(푋). (The proof below is a combination of the proofs in [Fel62] and [Pau07]).

Theorem 5.1.2. If 푋 is a locally compact space, then ℱ(푋) with the Chabauty topology is a compact Hausdorff space.

Proof. The compactness follows from the above theorem, so we just need to show the Hausdorff condition. Let 퐹1, 퐹2 be two distinct elements of ℱ(푋). Suppose (WLOG) that 푥 ∈ 퐹1 ∖퐹2. Since

퐹2 is closed, it equals 퐹 2, and so 푥∈ / 퐹2 implies that there exists some open neighborhood 푉 of

푥 such that 푉 ∩ 퐹2 = ∅. By local compactness of 푋, there exists a compact neighborhood 퐾 of 푥 158

with 퐾 ⊂ 푉 . Thus 퐾 ∩ 퐹2 = ∅, hence 퐹2 ∈ 풪퐾 .

Now set 푈 = Int(퐾), the interior of 퐾. Then we have 퐹1 ∩ 푈 ̸= ∅, since both contain 푥, thus

′ 퐹1 ∈ 풪푈 . Clearly the open sets 풪퐾 and 풪푈 are disjoint since 푈 ⊂ 퐾. Thus, we have disjoint open sets in ℱ(푋) containing the disjoint closed sets (in 푋) 퐹1 and 퐹2, and so we may conclude that ℱ(푋) is Hausdorff.

Before we move on, let us note a few more interesting properties of the Chabauty topology [Fel62].

Proposition 5.1.3. Suppose 푋 is a locally compact space and ℱ(푋) is the space of closed subsets in 푋 with the Chabauty topology. Then

1. The operation of union is continuous with respect to the Chabauty topology:

⋃︁ (−, −): ℱ(푋) × ℱ(푋) → ℱ(푋) ⋃︁ (퐴, 퐵) ↦→ 퐴 퐵

2. Let 푌 be a closed subset of 푋. Then the Chabauty topology on ℱ(푌 ) is equivalent to the subspace topology on ℱ(푌 ) inherited from ℱ(푋).

Proof. (1) The union of two closed sets is again closed, so the map ∪(−, −) is well-deﬁned. To show that the union map is continuous, we just need to show that the inverse image of any sub-basis element is open in ℱ(푋) × ℱ(푋).

So let 풪퐾 be a sub-basis element, where 퐾 denotes a compact subset of 푋. The inverse image of 풪퐾 under the union map is the set

퐴 = {(퐸, 퐹 ) ∈ ℱ(푋) × ℱ(푋) | 퐸 ∪ 퐹 ∈ 풪퐾 }. 159

Let (퐸, 퐹 ) ∈ 퐴. By deﬁnition, 퐸 ∪ 퐹 ∈ 풪퐾 and hence:

(퐸 ∪ 퐹 ) ∩ 퐾 = ∅ ⇐⇒ (퐸 ∩ 퐾) ∪ (퐹 ∩ 퐾) = ∅

⇐⇒ 퐸 ∩ 퐾 = ∅ and 퐹 ∩ 퐾 = ∅

⇐⇒ 퐸 ∈ 풪퐾 and 퐹 ∈ 풪퐾 .

Thus, (퐸, 퐹 ) ∈ 풪퐾 × 풪퐾 = 퐴 which shows that 퐴 is open in ℱ(푋) × ℱ(푋).

′ Now consider the other type of sub-basis element, 풪푈 , where 푈 denotes a non-empty open

′ subset of 푋. Denote the inverse image of 풪푈 under the union map by

′ 퐵 = {(퐸, 퐹 ) | 퐸 ∪ 퐹 ∈ 풪푈 }.

Let (퐸, 퐹 ) ∈ 퐵. Then it follows that either 퐸 ∩ 푈 ̸= ∅ or 퐹 ∩ 푈 ̸= ∅. Then the point (퐸, 퐹 ) lies

′ ′ in 풪푈 × ℱ(푋) or lies in ℱ(푋) × 풪푈 . Regardless, either set lies in 퐵. Indeed, suppose WLOG

′ that we have (퐶, 퐷) ∈ 풪푈 × ℱ(푋). Then

(퐶 ∪ 퐷) ∩ 푈 = (퐶 ∩ 푈) ∪ (퐷 ∩ 푈) ̸= ∅

′ since 퐶 ∈ 풪푈 . Hence 퐵 is open. This proves (1). (2) Let 푌 ⊂ 푋 be closed. We naturally endow 푌 with the subspace topology inherited from 푋. Then anything closed in 푌 is also closed in 푋 and ℱ(푌 ) ⊂ ℱ(푋). The subspace topology on ℱ(푌 ) inherited from ℱ(푋) can be generated by subbasis elements

풪퐾 ∩ ℱ(푌 ) = {퐹 ∩ 푌 | (퐹 ∩ 푌 ) ∩ 퐾 = ∅},

′ 풪푈 ∩ ℱ(푌 ) = {퐹 ∩ 푌 | (퐹 ∩ 푌 ) ∩ 푈 ̸= ∅}, where 퐾 is compact in 푋 and 푈 is a non-empty open set in 푋. The Chabauty space of ℱ(푌 ) is 160 formed by subbasis sets:

풫퐶 = {퐸 ∈ ℱ(푌 ) | 퐸 ∩ 퐶 = ∅},

′ 풫푉 = {퐸 ∈ ℱ(푌 ) | 퐸 ∩ 푉 ̸= ∅}, where 퐶 is compact in 푌 and 푉 is a non-empty open set in 푌 . By deﬁnition of the subspace topology on 푌 , any 퐸 ∈ ℱ(푌 ) is of the form 퐸 = 퐹 ∩ 푌 , where 퐹 is closed in 푋. Similarly, any 푉 open in 푌 is of the form 푉 = 푈 ∩ 푌 , where 푈 is open in 푋. In fact, all open sets in 푌 are formed in this way. Hence, we can re-write the above subbasis sets as:

풫퐶 = {퐹 ∩ 푌 | 퐹 ∈ ℱ(푋) and (퐹 ∩ 푌 ) ∩ 퐶 = ∅},

′ 풫푉 = {퐹 ∩ 푌 | 퐹 ∈ ℱ(푋) and (퐹 ∩ 푌 ) ∩ 푈 ̸= ∅}, since (퐹 ∩ 푌 ) ∩ (푈 ∩ 푌 ) = (퐹 ∩ 푌 ) ∩ 푈. Since any 퐶 compact in 푌 is also compact in 푋,

′ ′ we see that 풫퐶 = 풪퐶 ∩ ℱ(푌 ) and 풫푉 = 풪푈 ∩ ℱ(푌 ). Thus, ℱ(푌 ) with the Chabauty topology has the same subbasis sets as ℱ(푌 ) with the subspace topology and therefore these topologies are equivalent.

5.2 Chabauty Topology Metrizability

In this section, we discuss several properties of the Chabauty topology on ℱ(푋) regarding metrization. Metric spaces are some of the most well-behaved of topological spaces, so it is worth knowing when and how ℱ(푋) can be endowed with a metric. Our ﬁrst result shows that, if the underlying space 푋 is already a metric space, we all but have ℱ(푋) a metric. We just need 푋 is also locally compact. (cf. [Pau07, Prop. 1.8])

Proposition 5.2.1. Suppose (푋, 푑) is a locally compact metric space. Then the Chabauty topology 161 on ℱ(푋) is equivalent to the metric topology induced by the following metric:

⎧ ⃒ ⎫ ⃒ 푐 푐 ⎪ ⃒ 퐹1 ∪ 퐵 (*, 1/휀) ⊂ 풱휀(퐹2 ∪ 퐵 (*, 1/휀)) ⎪ ⎪ ⃒ ⎪ ⎨ ⃒ ⎬ 푑Hau(퐹1, 퐹2) = inf 휀 > 0 ⃒ ⃒ ⎪ ⃒ ⎪ ⎪ ⃒ 푐 푐 ⎪ ⎩ ⃒ 퐹2 ∪ 퐵 (*, 1/휀) ⊂ 풱휀(퐹1 ∪ 퐵 (*, 1/휀)) ⎭

where 풱휀 stands for the 휀-neighborhood: for 푆 ⊂ 푋,

풱휀(푆) = {푥 ∈ 푋 | 푑(푥, 푆) < 휀};

and 퐵(*, 1/휀) is the ball of radius 1/휀 about an arbitrarily chosen base point in 푋.

Remark 5.2.2. The above metric is referred to as pointed Hausdorff distance. One can easily show that ⋃︁ 풱휀(푆) = 퐵(푠, 휀). 푠∈푆

Note also the similarity of 푑Hau with the Hausdorff distance: for 퐴, 퐵 ⊂ 푋,

푑퐻 (퐴, 퐵) = inf{휀 > 0 | 퐴 ⊂ 풱휀(퐵) and 퐵 ⊂ 풱휀(퐴)}

On the collection of all subsets of 푋, this forms a semi-metric. If we let ℱ푏(푋) denote the collec-

tion of all closed and bounded subsets of 푋, then (ℱ푏(푋), 푑퐻 ) forms a metric. If the space 푋 is compact, then ℱ(푋) equipped with 푑퐻 is a compact metric space (see [BH10, I.5]). The following is stated in [CEG87, Ch. I.3] and we prove it below.

Proposition 5.2.3. If 푋 is compact and metrizable, then the Chabauty topology on ℱ(푋) agrees with the topology induced by the Hausdorff metric.

Proof. Since 푋 is compact and metrizable, we have that closed sets are compact and thus bounded.

Thus ℱ(푋) = ℱ푏(푋) in this case. Now we must show that the Chabauty topology on ℱ(푋) is equivalent to the Hausdorff metric. 162

Let 퐹 be a closed set in 푋. Consider an open ball 퐵푟(퐹 ) in ℱ(푋) with the Hausdorff distance.

We must ﬁnd a basis element ℬ(퐾, 풰) that contains 퐹 and is contained in 퐵푟(퐹 ).

If 퐸 ∈ 퐵푟(퐹 ), then we have that 퐹 ⊂ 풱푟(퐸) and 퐸 ⊂ 풱푟(퐹 ). Let 퐾 = 푋 ∖ 풱푟(퐹 ). This is a

compact set and clearly 퐹 ∈ 풪퐾 . If 퐸 ∈ 풪퐾 , then necessarily we have 퐸 ⊂ 풱푟(퐹 ). Now because 퐹 is compact, it can be covered by ﬁnitely many balls of radius 푟, say the collection

풰 = {퐵1, 퐵2, . . . , 퐵푛}.

′ Suppose 퐸 ∈ 풪풰 . Then 퐸 intersects every ball in 풰. If 푥 ∈ 퐹 , then 푥 ∈ 퐵푗 for some 푗, and hence

for some 푦 ∈ 퐸, 푑(푥, 푦) < 푟. Therefore, 푑(푥, 퐸) < 푟 and we have that 퐹 ⊂ 풱푟(퐸). Thus, we have

퐹 ∈ ℬ(퐾, 풰) ⊂ 퐵푟(퐹 ). This shows that the Chabauty topology is finer than the Hausdorff metric. Now consider a closed set 퐹 in 푋 and a basis element ℬ(퐾, 풰) containing 퐹 . To show that the Hausdorff metric is finer than the Chabauty topology, we must find an open ball containing 퐹 that lies in ℬ(퐾, 풰). We are given that 퐹 and 퐾 are disjoint. Hence, in ℱ(푋) we can find disjoint open

balls B퐾 and B퐹 containing 퐾 and 퐹 , respectively, in the Hausdorff metric. If 퐸 is any closed

set in 푋 with 퐸 ∈ B퐹 , then clearly 퐸 ∈ ℬ(퐾, 풰), since B퐹 is disjoint from 퐾, and any open set

from 풰 that intersects 퐹 will necessarily intersect B퐹 , hence 퐸 also. We have just shown that the Hausdorff metric is ﬁner than the Chabauty topology on ℱ(푋). Since each topology is ﬁner than the other, we must have that they are equivalent.

We can also describe the convergence of sequences in ℱ(푥) via the criterion below. In light of the above proposition, this gives us another characterization of the topology when 푋 is a locally compact metric space.

Proposition 5.2.4. Suppose 푋 is a locally compact metric space, let (퐹푛)푛∈ℕ be a sequence in ℱ(푋), and 퐹 ∈ ℱ(푋). Then the following are equivalent:

1. (퐹푛) converges to 퐹 in the Chabauty topology.

2. The sequence (퐹푛)푛∈ℕ and 퐹 satisfy the following: 163

(a) For all 푥 ∈ 퐹 , there exists a sequence (푥푛)푛∈ℕ converging to 푥 in 푋 such that, for

every 푛, 푥푛 ∈ 퐹푛.

(b) For all strictly increasing sequences (푖푗)푗≥1 and for all sequences (푥푖푗 )푗≥1 in 푋 such

that 푥푖푗 ∈ 퐹푖푗 and 푥푖푗 → 푥 in 푋, we have 푥 ∈ 퐹 .

Proof. We ﬁrst prove (1) ⇒ (2). Suppose (퐹푛) converges to 퐹 in the Chabauty topology. This

means, for any basis neighborhood ℬ(퐾, 풰) of 퐹 , there exists an integer 푁 > 0 such that 퐹푛 ∈ ℬ(퐾, 풰) for all 푛 ≥ 푁.

1 Now ﬁx 푥 ∈ 퐹 . For each positive integer 푖, let 퐵푖 denote the ball of radius 푖 centered about 푥 in 푋. Note that the collection {퐵푖}푖≥1 forms a nested neighborhood basis of 푥. Now we have, for 푖 ≥ 1 퐹 풪′ 푖 푁 = 푁(푖) each , a neighborhood of , namely 퐵푖 . Thus, for every , there exists an integer 푖 such that 퐹푗 ∩ 퐵푖 ̸= ∅ for all 푗 ≥ 푁(푖). We can now construct a sequence (푥푗)푗≥1 in 푋 for

푖 every given integer 푖 ≥ 1. For 푗 = 1, 2, . . . , 푁(푖) − 1, choose 푥푗 ∈ 퐹푗. For 푗 ≥ 푁(푖), choose

푖 푥푗 ∈ 퐹푗 ∩ 퐵푖.

Now we shall use a diagonalization argument to construct a sequence (푥푚)푚∈ℕ in 푋 converging

푚 to the given point 푥 ∈ 퐹 , and with each 푥푚 ∈ 퐹푚 for all 푚. For each 푚 ∈ ℕ, let 푥푚 = 푥푚. By the

푖 construction above, the point 푥푖 ∈ 퐹푖 ∩ 퐵푖 for every 푖 ≥ 1. Hence 푥푚 ∈ 퐹푚 for all 푚. To show

(푥푚) converges to 푥 in 푋, let 푈 be a neighborhood of 푥. Choose 퐾 large enough so that 퐵퐾 ⊂ 푈.

Then there exists 푁(퐾) such that 퐹푗 ∩ 퐵퐾 ̸= ∅ for all 푗 ≥ 푁(퐾). Now set 푀 = max(퐾, 푁(퐾)). Then for any 푗 ≥ 푀, we have

푗 푥푗 = 푥푗 ∈ 퐹푗 ∩ 퐵푗 ⊂ 퐹푗 ∩ 퐵퐾 ⊂ 퐹푗 ∩ 푈 ⊂ 푈

Thus the sequence (푥푚) converges to 푥, proving (a).

To show (b), we proceed by contradiction. Suppose we have a sequence (푥푖푗 ) as given above,

but assume 푥 ∈ 푋 ∖ 퐹 . Because 퐹 is closed, there exists an open neighborhood 푉 of 푥 such that 푉 ∩ 퐹 = ∅. By the local compactness of 푋, we have that there exists a compact neighborhood 퐾 of 푥 such that 퐾 ⊂ 푉 . Thus, the set 풪퐾 is a neighborhood of 퐹 in the Chabauty topology. So 164

there exists some 푁 > 0 such that, if 푛 ≥ 푁, then 퐹푛 ∈ 풪퐾 , meaning 퐹푛 ∩ 퐾 = ∅. But then we also have 퐹푛 ∩ 푉 = ∅ for all 푛 ≥ 푁 as well. Thus, we have found a neighborhood 푉 of 푥 such

that, eventually, no point 푥푖푗 lies in 푉 . This contradicts our assumption that (푥푖푗 ) converges to 푥, and therefore 푥 must lie in 퐹 . Now we show (2) ⇒ (1). Assume (a) and (b) are true. Let ℬ(퐾, 풰) be a neighborhood of 퐹 in

ℱ(푋), where 퐾 is a compact set in 푋, and 풰 = {푈1, . . . , 푈푚} is a collection of non-empty open sets in 푋. Then 퐹 ∩ 퐾 = ∅ and 퐹 ∩ 푈푖 ̸= ∅ for each 푖 = 1, . . . , 푚. Choose points 푦푖 ∈ 퐹 ∩ 푈푖. For

푖 each point 푦푖, we have by (a) that there exists a sequence (푦푗)푗≥1 converging to 푦푖 in 푋, and such

푖 that 푦푗 ∈ 퐹푗 for all 푗. Since each set 푈푖 is an open set containing 푦푖, we have that convergence of

푖 푖 (푦푗) implies there exists an integer 푁(푖) such that, for 푗 ≥ 푁(푖), 푦푗 ∈ 푈푖. Hence, 퐹푗 ∩ 푈푖 ̸= ∅ for 푗 ≥ 푁(푖).

This almost gives us that (퐹푛) converges to 퐹 , but we must make sure that 퐹푛 lies in 풪퐾 for all 푗 ≥ 푁(푖). To do so, we will prove something slightly more general. Consider any subsequence

(퐹푖푗 )푗≥1 of (퐹푛) such that 퐹푖푗 ∩ 퐾 ̸= ∅ for all 푗. Then we can choose a sequence of points in 푋, say 푥푖푗 ∈ 퐹푖푗 ∩ 퐾 for each 푗, giving us a sequence (푥푖푗 ) lying in 퐾. By compactness of 퐾, we may assume, after possibly passing to a subsequence, that this sequence converges to some point 푥. However, since 푋 is metrizable, it is Hausdorff, and therefore 퐾 compact means 퐾 is also closed. Thus 푥 ∈ 퐾. But by (b), we must have 푥 ∈ 퐹 , thus contradicting 퐹 ∩ 퐾 = ∅. So it cannot be the case that such a subsequence exists. Applying this to the above situation, we can immediately see that, if 푗 ≥ 푁, then we must have 퐹푗 ∩퐾 = ∅, else we could construct a contradictory subsequence as above. Therefore, we have 퐹푗 ∈ ℬ(퐾, 풰) for all 푗 ≥ 푁, and thus (퐹푛) converges to 퐹 in the Chabauty topology.

Remark 5.2.5. The above proof is essentially taken from [Pau07, Proposition 1.8], which is in French. Moreover, a proof can be found in several other places. For instance, in [GR06] where the convergence criterion (2) in the proposition is referred to as geometric convergence. As far as [dlH08] knows, the word “geometric” is ﬁrst associated to the Chabauty topology in [Thu80, 165 Chapter 9], where it is called the geometric topology. See also [CEG87, Ch. I.3], or the recent reprint [CEG06].

Here is another interesting result regarding the metrizability of ℱ(푋), which comes from [CEG87], [CEG06]:

Proposition 5.2.6. Suppose 푋 is a locally compact, Hausdorff, and second countable. Then ℱ(푋) with the Chabauty topology is second countable and metrizable.

Proof. We showed above that ℱ(푋) is always compact, no matter the topological space 푋. It is also Hausdorff since 푋 is locally compact. Moreover, every compact Hausdorff space is normal, that is, ℱ(푋) is a 푇4 space. Since 푇4 ⇒ 푇3, we have that ℱ(푋) is also a regular space. If we can show that ℱ(푋) contains a countable basis, then by the Urysohn Metrization Theorem, ℱ(푋) will be metrizable.

Since 푋 is second countable, it has a countable basis, say {퐵푛}푛∈ℕ. Because 푋 is Hausdorff and locally compact, 푋 is also regular and hence we may assume that each basis element has

′ compact closure. Let us show that the collection {풪퐵푛 , 풪퐵푛 }푛≥1 of Chabauty subbasis elements forms a countable basis for the Chabauty topology.

′ ′ First, consider a subbasis element 풪푈 for some open set 푈 in 푋. Then for 퐹 ∈ 풪푈 , we have that there is some element 푥 ∈ 퐹 ∩ 푈. There necessarily exists a basis element 퐵푛 containing 푥

′ ′ with 퐵푛 ⊂ 푈. Hence, 퐹 ∈ 풪퐵푛 ⊂ 풪푈 .

Now consider a basis element 풪퐾 for some compact set 퐾 in 푋. Let 퐹 ∈ 풪퐾 . By regularity of

푋, for every 푥 ∈ 퐾, we have a basis element 퐵푛(푥) containing 푥 and disjoint from 퐹 . Moreover,

we have that 퐵푛(푥) is also disjoint from 퐹 . Thus, {퐵푛(푥)}푥∈퐾 forms an open cover of 퐾 and by compactness of 퐾 we have that 퐾 can be covered by only ﬁnitely many of these basis elements.

Denote these elements by 퐵푛(푖) for 푖 = 1, 2, . . . , 푘. Then we see that

푘 ⋂︁ 퐹 ∈ 풪 ⊂ 풪 . 퐵푛(푖) 퐾 푖=1 166 ′ This proves that the collection {풪퐵푛 , 풪퐵푛 }푛≥1 forms a countable basis for the Chabauty topology, and hence ℱ(푋) is metrizable by the Urysohn Metrization Theorem.

Remark 5.2.7. Note that the above proposition is not really a generalization of Proposition 5.2.1. This is because every locally compact Hausdorff space is regular. So the Urysohn Metrization Theorem also yields that the underlying space 푋 is metrizable, in which case Proposition 5.2.1 applies and we have ℱ(푋) is metrizable.

The following is cited from [dlH08], but the original proof is from [Fel62]. Recall that a topological space 푋 admits a one-point compactiﬁcation 푌 = 푋 ⊔ {∞}, unique up to homeomorphism, if and only if 푋 is locally compact and Hausdorff. In this situation, we have that ℱ(푋) is homeomorphic to a subspace of ℱ(푌 ):

Proposition 5.2.8. Suppose 푌 = 푋 ⊔{∞} is the one-point compactiﬁcation of 푋. Deﬁne the map 휑 : ℱ(푋) → ℱ(푌 ) by: 휑(퐹 ) = 퐹 ⊔ {∞} for 퐹 a closed subset of 푋. Then 휑 is a homeomorphism from ℱ(푋) to the subspace 푊 = {퐶 ∈ ℱ(푌 ) | 퐶 ⊃ {∞}} of ℱ(푌 ). Moreover, the subset 푍 = {{푥} ∈ ℱ(푋) | 푥 ∈ 푋} and its image 휑(푍) are homeomorphic to 푌 .

5.3 The Space of Closed Subgroups - Chabauty Space

Let us now spice things up and imagine we are in the world of topological groups. If 퐺 is a locally compact topological group, then we have all the relevant results above for the space of closed subsets, ℱ(퐺). But since we are dealing now with a group structure, one would naturally like to study instead the set of all closed subgroups of 퐺. We will denote this set by 풞(퐺). Clearly we have the inclusion 풞(퐺) ⊂ ℱ(퐺). The following proposition tells us that we don’t lose any of the above nice topological properties by focusing on 풞(퐺).

Proposition 5.3.1 ([BdlHK09], [Hae11], [Fel62], [dlH08]). Let 퐺 be a locally compact group. Equip ℱ(퐺) with the Chabauty topology and let 풞(퐺) denote the subset of ℱ(퐺) of all closed subgroups of 퐺. Then 167 1. The subspace 풞(퐺) is closed in ℱ(퐺); so in particular, it is compact.

2. For any 퐻 ∈ 풞(퐺), the collection of subsets of the form

풩퐾,푈 (퐻) = {퐶 ∈ 풞(퐺) | 퐶 ∩ 퐾 ⊂ 퐻푈 and 퐻 ∩ 퐾 ⊂ 퐶푈}

where 퐾 ⊂ 퐺 is compact and 푈 ⊂ 퐺 is non-empty and open, forms a basis of neighborhoods at 퐻.

Deﬁnition 5.3.2. The space of closed subgroups 풞(퐺) of a locally compact group 퐺 is called the Chabauty Space of 퐺 when equipped with the Chabauty topology.

It was in studying closed subgroups of locally compact topological groups that Chabauty [Cha50] ﬁrst introduced this topology. Also Harvey [Har77] utilized this topology in studying Fuschian groups. In [CEG87], [CEG06] the Chabauty topology is used to derive some results about spaces of hyperpolic manifolds. We will not get into any of that here, but there is an interesting result concerning the Chabauty space for a Lie group, which includes Chabauty’s original result [Cha50] but for Lie groups.

Theorem 5.3.3. [CEG06, I.1.3.1.4] Let 풞(퐿) be the space of closed subgroups of a Lie group 퐿 with the Chabauty topology (so 풞(퐿) is compact and metrizable). Let 푈 be an open neighborhood of the trivial subgroup {e} in 퐿. Then:

1. 퐴(푈) = {퐺 ∈ 풞(퐿) | 퐺 ∩ 푈 = {푒}} is compact.

2. 퐵(푈) = {퐺 ∈ 퐴(푈) | 퐺 is torsion-free} is compact.

3. The set of discrete subgroups of 퐿 is open in 풞(퐿) and equals the union of the interiors in 풞(퐿) of the compact spaces 퐴(푈), as 푈 varies over open neighborhoods of {푒} in 퐿.

4. If {Γ(푛)} is a sequence of discrete subgroups converging to the discrete subgroup Γ, then

휇Γ ≤ lim inf 휇Γ(푛). 푛→∞ 168 In particular, 퐴(푈, 푀) = {퐺 ∈ 퐴(푈) | 휇(퐺) ≤ 푀} and 퐵(푈, 푀) = {퐺 ∈ 퐵(푈) | 휇(퐺) ≤ 푀 are compact.

Examples

There are in fact very few explicit examples of an easily understood Chabauty space. Below we list most of the known examples.

Example 5.3.4. Let 퐺 = ℝ. The Chabauty space is homeomorphic to a compact interval: 풞(ℝ) ≃ [0, ∞]. The homeomorphism is given by:

0 ←→ {0} 1 휆 ←→ , where 0 < 휆 < ∞ 휆ℤ ∞ ←→ ℝ

This is one of the simplest examples.

Example 5.3.5. Another simple example is 풞(ℤ). This space is homeomorphic to the subspace 1 { 푛 }푛≥1 ∪ {0} of [0, 1]. This map is given by:

1 ←→ 푛 푛 ℤ 0 ←→ {0}

Note that 풞(ℤ) is homeomorphic to the one-point compactiﬁcation of ℤ+ (see Proposition 5.2.8).

1 Proof of Example 5.3.5. Let us show a proof for Example 5.3.5 that 풞(ℤ) ≃ 푋ℤ := {0}∪{ 푛 | 푛 ≥

1}. Denote by 휑ℤ the map from 푋ℤ → 풞(ℤ) deﬁned by:

(︂ 1 )︂ 휑 = 푛 ℤ 푛 ℤ

휑ℤ(0) = {0} 169 It is very easy to see that 푛ℤ and {0} are the only closed subgroups of ℤ, and thus this map is clearly bijective. To show that 풞( ) ≃ 푋 , we must show that 휑 and 휑−1 are continuous. ℤ ℤ ℤ ℤ

First we show that 휑ℤ is continuous. Note that for 푛ℤ ∈ 풞(ℤ), we have that:

{푛ℤ} = ℬ({푛}, {1, 2, . . . , 푛 − 1}),

and hence any open neighborhood of 푛ℤ in 풞(ℤ) must contain {푛ℤ}. So to show that 휑ℤ is 1 1 continuous at each 푛 ∈ 푋ℤ, it sufﬁces to show that there exists a neighborhood 푉 of 푛 in 푋ℤ such 1 1 1 that 휑ℤ( 푛 ) ⊂ {푛ℤ}. But clearly { 푛 } is open in 푋ℤ, and thus setting 푉 = { 푛 } will do.

Now we must show that 휑ℤ is continuous at 0 in 푋ℤ. Let ℬ(풰, 퐾) be any basis neighborhood of {0} in 풞(ℤ). If we can show that, for some integer 푁 ≥ 1, 푛ℤ ∈ ℬ(풰, 퐾) for all 푛 ≥ 푁, then 1 we’ll have that 푉 = {0} ∪ { 푛 | 푛 ≥ 푁} is a neighborhood of 0 in 푋ℤ with 휑ℤ(푉 ) ⊂ ℬ(풰, 퐾).

Hence, this would show continuity of 휑ℤ at 0. Since {0} ∈ ℬ(풰, 퐾), we have {0} ∩ 푈 ̸= ∅ for each 푈 ∈ 풰; hence 0 ∈ 푈 for each 푈 ∈ 풰.

Also, since {0} ∩ 퐾 = ∅, we have 0 ∈/ 퐾. Because 퐾 is compact in the discrete space ℤ, we have

that 퐾 must be ﬁnite. Set 푁 = 1 + max푘∈퐾 {푘}. Then for all 푛 ≥ 푁, we’ll have 푛ℤ ∩ 푈 ̸= ∅ for each 푈 ∈ 풰; and 푛ℤ ∩ 퐾 = ∅ since 0 ∈/ 퐾 and all non-zero elements of 푛ℤ are greater than any element of 퐾, by our choice of 푁. Lastly, note that by exactly the same argument above, we can show that 휑−1 is continuous at ℤ

{0} and 푛ℤ for each 푛 ≥ 1. Thus 휑ℤ is indeed a homeomorphism.

Remark 5.3.6. In showing that 휑ℤ was a homeomorphism, we used the fact that the topology on ℤ is discrete. More generally, if a space 푋 is a discrete, locally compact metric space, then part of Proposition 5.2.4 can be simpliﬁed:

Lemma 5.3.7. With the above hypotheses, suppose {퐹푛} is a sequence in ℱ(푋) converging to 퐹 .

Then for all 푥 ∈ 퐹 , there exists 푁푥 > 0 such that 푥 ∈ 퐹푛 for all 푛 ≥ 푁푥.

Proof. By the equivalence given in Proposition 5.2.4, we know that, for each 푥 ∈ 퐹 , there exists 170

a sequence (푦푛(푥))푛≥1 in 푋 converging to 푥 in 푋 such that 푦푛(푥) ∈ 퐹푛 for all 푛. But since 푋 is discrete, 푦푛(푥) → 푥 means, for some 푁푥, we have 푦푛(푥) = 푥 for all 푛 ≥ 푁푥. Therefore, we must

have 푥 ∈ 퐹푛 for all 푛 ≥ 푁푥.

Recall that, when considering a topology to place on a group, one can easily employ the discrete topology as this trivially makes a topological group. As the examples above illustrate, we often

would like to consider topological groups 퐺 that are locally compact metric spaces, such as ℝ and ℤ. So it is interesting to note that, if such a group 퐺 is countable, then the topology on 퐺 must be the discrete topology. More precisely, we have the following proposition:

Proposition 5.3.8. Let 퐺 be a countable, locally compact, Hausdorff group. Then the topology on 퐺 is the discrete topology.

Proof. Assume that the topology on 퐺 is not discrete. Then there must exist at least one point, say

푥0 ∈ 퐺, such that the singleton {푥} is not open in 퐺. In other words, for any open neighborhood

푈 of 푥0, there exists a point 푦푈 ̸= 푥0 in 푈. But this means 푥0 is a limit point in 퐺. Then by the homogeneity of 퐺, we must have that every point of 퐺 is a limit point. Indeed, for any 푦 ∈ 퐺, there exists a 푔 ∈ 퐺 such that 푔푥0 = 푦. Then for any neighborhood 푈 of 푥0, we have 푔푈 is a neighborhood of 푦; so 푦푈 ∈ 푈 with 푦푈 ̸= 푥0 implies that 푔푦푈 ∈ 푔푈 and 푔푦푈 ̸= 푦.

Now for each 푥 ∈ 퐺, the set 퐶푥 = 퐺 ∖ {푥} is open, because singletons in 퐺 are closed (퐺 is

Hausdorff). Because 푥 is a limit point of 퐺, the closure of 퐶푥 is all of 퐺:

⋃︁ 퐶푥 = 퐺 ∖ {푥} {푥} = 퐺

Hence, each open set 퐶푥 is dense in 퐺. Recall that every locally compact Hausdorff space is also a

Baire space, meaning that any countable collection {푈푛} of open sets in 퐺, all of which are dense

in 퐺, must have dense intersection. Since 퐺 is countable, the collection {퐶푥}푥∈퐺 is a countable 171 collection of dense open sets. Therefore,

⋂︁ 퐺 = 퐶푥 푥∈퐺 ⋂︁ = 퐺 ∖ {푥} 푥∈퐺

= 퐺 ∖ 퐺

= ∅ which is obviously a contradiction (we are assuming 퐺 is non-empty, because otherwise there would be nothing to show).

Example 5.3.9. Exercise[dlH08]: Determine 풞(SO(3)) and 풞(aff(ℝ)). One could also show that 풞(ℝ/ℤ) and 풞(ℤ) are homeomorphic, but this follows trivially from Pontryagin Duality ([Cor11]). In [BdlHK09, Sec. 2], it is shown as a “warm-up” that 풞(aff(ℝ)) is a compact and contractible space of dimension 2. Recall that

⎧⎛ ⎞ ⎫ ⎨⎪ 푒푥 푦 ⎬⎪ ⎜ ⎟ aff(ℝ) = ⎝ ⎠ : 푥, 푦 ∈ ℝ ⎩⎪ 0 1 ⎭⎪

and is, up-to ismorphism, the unique non-abelian, connected, real Lie group of dimension 2.

Example 5.3.10. Let 퐺 = ℂ ≃ ℝ2. Then the Chabauty space 풞(ℂ) is homeomorphic to a 4- sphere. Specifically, it is the disjoint union of a 2-sphere and the product of an open interval with the complement of a trefoil knot in a 3-sphere. The main difficulty is figuring out how the “strata”

of 풞(ℂ) are glued together. The speciﬁcs of this are given in a paper by Hubbard and Pourezza [PH79]. See also the summary in [dlH08]. The 2-sphere piece corresponds to the space of closed

subgroups which are not lattices in 풞(ℂ), while the product (0, 1) × 푆3 ∖ 푇 corresponds to the subspace in 풞(ℂ) of lattices.

Example 5.3.11. The paper [BdlHK09] gives us another example, albeit far more complicated 172 then those above. This concerns the Chabauty space for the three dimensional Heisenberg group. They show that this space is 6-dimensional, path-connected, but not locally connected. For a more introductory summary, see [dlH08].

Example 5.3.12. Here is yet another example showcasing how difﬁcult it can be to describe the

Chabauty space for a particular topological group. Haettel completely describes in [Hae10b] 풞(ℝ× ℤ), and shows that this space is very non-trivial. For example, Haettel shows that its fundamental group contains the fundamental group of the Hawaiian Earrings, thus making the fundamental group of 풞(ℝ × ℤ) uncountable.

Chabauty Space - Recent Research Results

Above we mentioned a few examples of the Chabauty space 풞(퐺), a few of which are very recent results. In this section, I give a brief, chronological synopsis of other results related to the Chabauty space of a group.

The Chabauty space of ℝ푛 is simply connected

The results by Hubbard and Pourezza [PH79] on 풞(ℝ2) show that, even for a simple space like ℝ2, determining the structure of closed subgroups is difﬁcult. In general ℝ푛, note that any closed subgroup is isomorphic to ℝ푎 ⊕ ℤ푏, for some non-negative integers 0 ≤ 푎 + 푏 ≤ 푛 ([dlH08]).

For topological groups 퐻, 퐴, . . . , 퐵, we deﬁne 풞퐴,...,퐵(퐻) to be the subspace of 풞(퐻) of closed subgroups that are topologically isomorphic to one of 퐴, . . . , 퐵. Then given a pair of non-negative

푛 integers (푎, 푏) as above, we have that 풞ℝ푎⊕ℤ푏 (ℝ ) is a homogeneous space of GL푛(ℝ). For ex- 푛 푛 ample, 풞ℤ푛 (ℝ ) = ℒ(ℝ ) = GL푛(ℝ)/ GL푛(ℤ). The point here is that, identifying the closed subgroups is not the hard part - figuring out how all the “pieces” of 풞(ℝ푛) fit together is what is really difficult. In [Klo09], Kloeckner investigates the Chabauty space structure for higher-dimensional Eu-

clidean space and tries to determine how one could “glue” together the “pieces” of 풞(ℝ푛). Kloeck- ner does not achieve a complete, global description of 풞(ℝ푛) for 푛 > 2, but is able to describe locally what is going on. There are two main theorems. 173 Theorem 5.3.13. [Klo09, Theorem 1.2] For all 푛, the Chabauty space of ℝ푛 admits a Goresky- MacPherson stratiﬁcation. If 푛 ≥ 2, it is moreover a pseudo-manifold.

So in particular, 풞(ℝ푛) is not (homemorphic to) a manifold for 푛 > 2. This stratification means that 풞(ℝ푛) has the structure of a gluing of manifolds, done in a “nice” way. For some introductory definitions and examples on stratified spaces, see [BH10, Chapter II.12]. The other main result is:

Theorem 5.3.14. [Klo09, Theorem 1.3] For all 푛, the Chabauty space of ℝ푛 is simply connected.

Compare this to the results by [Hae10b] on 풞(ℝ × ℤ), which has an uncountable fundamental group.

Chabauty Compactiﬁcation

In the last 7 years, much work has been done by the French mathematician Thomas Haettel regarding Chabauty space [Hae10b], [Hae10a], [Hae13b], [Hae13a]; or see his Ph.D thesis [Hae11]. As mentioned above, in [Hae10b], Haettel determines the structure of the Chabauty space

풞(ℝ × ℤ). He identiﬁes part of 풞(ℝ × ℤ) with the space known as the “Hawaiian Earrings,” a subspace 푋 ⊂ ℝ2 deﬁned by ⋃︁ 푋 = 퐶푛 푛≥1

where 퐶푛 is a circle of radius 1/푛 and center (1/푛, 0). It is well known that 휋1(푋) is uncountable [Hat02, Example 1.25, p.49].

Another part of 풞(ℝ × ℤ) is comprised of a countable collection of “cones.” These cones are glued together with 푋 above to form 풞(ℝ × ℤ) [Hae10b, p. 1], [Hae10b, Figure 6, p. 18]. Haettel shows that 풞(ℝ × ℤ) is path-connected, but not locally connected. Note that this is similar to the results of [BdlHK09] where it is shown that 풞(ℋ) (ℋ = Heisenberg group) is a 6-dimensional path-connected space that is not locally connected. We recall from [BH10, II.10] that a symmetric space is a connected Riemannian manifold 푀

where, for each point 푝 ∈ 푀, there is an isometry 휎푝 of 푀 such that 휎푝(푝) = 푝 and the differential

of 휎푝 at 푝 is multiplication by −1. 174 If we have a symmetric space 푀 that is simply-connected and non-positively curved, and 푀

has a trivial Euclidean de Rham factor (푀 = 푁 × 피푛, 피푛 is the de Rham factor), then 푀 is said to be of non-compact type. In this case, the connected component of the identity in Isom(푀) is a semi-simple Lie group with trivial center and no compact factors. There is a correspondence between these types of Lie groups and symmetric spaces of non-compact type. Specifically, if one is given such a Lie group 퐺 and maximal compact subgroup 퐾, then the space 푀 = 퐺/퐾 can be endowed with a 퐺-invariant Riemannian metric, and is a symmetric space of non-compact type. Moreover, the connected component of Isom(푀) is 퐺 (see [BH10, II.10]). In [Hae10a], Haettel takes a symmetric space 푀 of non-compact type with isometry group 퐺 = Isom(푀). 푀 can be identified with the subspace of maximal compact subgroups of 퐺 by sending each point of 푀 to its stabilizer in 퐺 (퐺 acts continuously - by isometries - on 푀). That is, we have a map from 푋 to the Chabauty space of 퐺. The closure in 풞(퐺) of the image of 푋 is called the Chabauty compactification of 푋. Haettel describes the subgroups that appear in the boundary of this compactification. In [Hae13b], Haettel shows that the symmetric space associated to a rank one, real semi-simple Lie group 퐺, with finite center and without compact factor, has Chabauty compactification the set of all closed, connected, Abelian subgroups of dimension the real rank of 퐺, with real spectrum.

Haettel also shows this when 퐺 = SL3(ℝ) or 퐺 = SL4(ℝ).

5.4 Metrizability of Topological Groups

Very soon we will discuss the space of closed subgroups of 퐺 = Aut(푋), 푋 a locally ﬁnite tree. In particular, we want to study the space of lattices, which is contained in the space of closed subgroups. There is a natural topology for these spaces, which we study in the sections to follow. First, we wish to understand ways in which topological groups are metrizable. Our future goal is to place a metric on 퐺, which will induce a metric on the space of closed subgroups, allowing us to study lattices via this metric. Firstly, we might wish to know when a topological group is even metrizable. An existence theorem comes to us from Birkhoff and Kakutani: 175 Theorem 5.4.1 (Birkhoff-Kakutani). A topological group is metrizable if and only if there is a countable base of neighborhoods of the identity.

Proof for this theorem can be found in [HR79, (8.3)], [Dik13, (4.1.16)], or even [MZ55, I.22]. All require an application of a lemma similar to that given below. First, a deﬁnition:

Deﬁnition 5.4.2. A pseudo-norm on a group 퐺 is a map 푣 : 퐺 → ℝ+ such that

1. 푣(푒) = 0;

2. 푣(푥−1) = 푣(푥) for all 푥 ∈ 퐺;

3. 푣(푥푦) ≤ 푣(푥) + 푣(푦) for all 푥, 푦 ∈ 퐺.

If a pseudo-norm satisﬁes 푣(푥) = 0 if and only if 푥 = 푒, then it is called a norm.

One of course obtains a pseudo-metric or metric 푑푣 from a pseudo-norm/norm 푣 by setting

−1 푑푣(푥, 푦) = 푣(푥 푦). This pseudo-metric (or metric) is naturally left-invariant:

푑푣(푔푥, 푔푦) = 푑푣(푥, 푦)

for all 푔, 푥, 푦 ∈ 퐺. Conversely, every left-invariant pseudo-metric 푑 gives rise to a pseudo-norm 푣푑 by setting 푣푑(푔) = 푑(푔, 푒). And now for that lemma I promised:

Lemma 5.4.3. Let 퐺 be a topological group. Let {푈푛}푛∈ℕ be a collection of symmetric neighbor-

2 hoods of 푒 such that 푈푛+1 ⊂ 푈푛 for 푛 = 1, 2, 3,... . Then there exists a continuous, left invariant pseudo-metric 푑 on 퐺 such that

푈푛+1 ⊂ 퐵1/푛(푒) ⊂ 푈푛

−1 for every 푛. If in addition, 푥푈푛푥 = 푈푛 for every 푥 ∈ 퐺 and 푛 = 1, 2,..., then 푑 is also right invariant.

The proof of Theorem 5.4.1 takes the given countable collection of identity neighborhoods and forms a symmetric collection with the property in the lemma. One then obtains a pseudo-metric via the lemma, which turns out to be an actual metric. This last part utilizes the following corollary: 176 Corollary 5.4.4. Let {푈 } be as in the above lemma. Then the set 퐻 = ⋂︀ 푈 is a closed 푛 푛∈ℕ 푛∈ℕ 푛 subgroups of 퐺 and 퐻 = {푔 ∈ 퐺 | 푑(푔, 푒) = 0}.

So in particular, 푑 is a true metric if and only if 퐻 = {푒}.

The following lemmas from [Hof06] give us greater ﬂexibility in metrizing a group. One should notice similarities with what we have already stated.

Lemma 5.4.5 ([Hof06, Lemma 2.7]). Let 퐺 be a Hausdorff topological group. TFAE:

1. There exists a left invariant metric 푑 on 퐺 inducing the topology on 퐺.

2. There exists a continuous function ‖·‖ : 퐺 → ℝ+ such that:

(a) ‖푥‖ = 0 if and only if 푥 = 푒;

(b) ‖푥−1‖ = ‖푥‖ for all 푥 ∈ 퐺;

1 (d) For every neighborhood 푈 of 푒, there exists an 푛 ∈ ℕ such that, if ‖푔‖ < 푛 , then 푔 ∈ 푈.

3. There exists a function 휙 : 퐺 → [0, 1] such that:

(a) 휙(푒) = 0;

1 (b) For every neighborhood 푈 of 푒, there exists an 푛 ∈ ℕ such that, if 휙(푔) < 푛 , then 푔 ∈ 푈;

1 휙(푔푢) ≤ 휙(푔) + 푛

holds for all 푔 ∈ 퐺 and 푢 ∈ 푈. 177 If these conditions hold, then the three functions 푑, ‖·‖, and 휙 are related as follows:

‖푥‖ = 푑(푥, 푒)

휙(푥) = min{‖푥‖, 1}

푑(푥, 푦) = sup{|휙(푔푦) − 휙(푔푥)| : 푔 ∈ 퐺}

Lemma 5.4.6 ([Hof06, Lemma 2.9]). Let 퐺 be any topological group. The following condition is equivalent to the three conditions in the lemma above:

4. There exists a function 푈 : (0, ∞) → U푒, where U푒 is the set of subsets of 퐺 containing 푒, such that the following conditions are satisﬁed:

(a) For all 푟 > 1, 푈(푟) = 퐺;

⋃︀ (b) For all 푠 > 0, 푟<푠 푈(푟) = 푈(푠);

1 (c) For every neighborhood 푁 of 푒, there is an 푛 ∈ ℕ such that 푈( 푛 ) ⊂ 푁;

(d) For each 푛 ∈ ℕ, there exists a neighborhood 푁 of 푒 such that

(︂ 1 )︂ 푈(푟)푁 ⊂ 푈 푟 + 푛

holds for all 푟 > 0.

The function 휙 from condition (3) above and the function 푈(·) here are related by the following:

휙(푔) = inf{푟 ∈ (0, 1] | 푔 ∈ 푈(푟)}

푈(푟) = {푔 ∈ 퐺 | 푝(푔) < 푟}

To summarize, the above lemmas tell us that, in order to have a left invariant metric on a topological group 퐺, it sufﬁces to construct the continuous function ‖·‖, the bounded function 푝, 178 or a function 푈 yielding a particular family of identity neighborhoods. Upon inspection of the proofs in [HR79] and [MZ55] of Theorem 5.4.1 and its accompanying lemma, one can see that the above lemmas from [Hof06] are a parsing of those proofs into pieces of lemmas.

5.5 The Metric On 퐺 = Aut(푋)

Now we wish to utilize the various results from above to place a metric on the automorphism group of a locally finite tree. We will take the latter route, and construct a family of identity neighborhoods satisfying the properties in Lemma 5.4.6. From this, we will obtain the function 휙 described in Lemma 5.4.5, which will allow us to define a left invariant metric on 퐺. We first define a countable family of neighborhoods of the identity in 퐺. Throughout this section, 푥0 ∈ 푋 is a fixed vertex. For each 푛 ∈ ℤ, define

⎧ ⎪ ⎨⎪{푔 ∈ 퐺 | 푔(푥0) ∈ 퐵푛(푥0)} if 푛 ≥ 1 푈푛 = ⎪ ⎩⎪FixG(퐵|푛|(푥0)) if 푛 ≤ 0

0 Here, FixG(푆) for a subset 푆 ⊂ 푋 stands for the pointwise stabilizer of 푆 by elements of 퐺:

Notice that our family {푈푛}푛∈ℤ has the following inclusions:

· · · ⊂ 푈−2 ⊂ 푈−1 ⊂ 푈0 ⊂ 푈1 ⊂ 푈2 ⊂ ....

Lemma 5.5.1. Let {푈푛}푛∈ℤ be the family deﬁned above.

1. For every 푛 ∈ ℤ, 푈푛 is an open neighborhood of id퐺.

2. ⋂︀ 푈 = {id } and ⋃︀ 푈 = 퐺. 푛∈ℤ 푛 퐺 푛∈ℤ 푛

Proof. For (1), consider 푛 ≤ 0. Each such 푈푛 is a point-stabilizer, and hence is a subgroup of 퐺. Moreover,

푈푛 = FixG(퐵|푛|(푥0)) = 풮(id퐺, 퐵|푛|(푥0)) 179

and so each such 푈푛 is a sub-basis element, and thus is open in 퐺. For 푛 ≥ 1, we ﬁrst see that since

id퐺(푥0) = 푥0, we have id퐺 ∈ 퐵푛(푥0), and hence id퐺 ∈ 푈푛. Moreover, if 푔 ∈ 푈푛 and 푕 ∈ 푈0, then

푔 ∘ 푕(푥0) = 푔(푥0)

and hence 푔 ∘ 푕 ∈ 푈푛. Thus we have 푔푈0 ⊂ 푈푛. Since the group operation in 퐺 is continuous, and we already know 푈0 is an open neighborhood, 푔푈0 is open as well. Therefore, we have 푈푛 is an open neighborhood for every 푛 ≥ 1. And this proves (1). For (2), note that since every 푔 ∈ 퐺 is either elliptic or hyperbolic, any elliptic 푔 will have its ∘ ﬁxed subtree 푇푔 ⊂ 퐵2푘(푥0) for some 푘 ≥ 0, and hence 푔 ∈ 푈2푘. For any hyperbolic 푔, we likewise → will have 푔 ∈ 푈푘 for some 푘 ≥ 1. Speciﬁcally, we have 푔 ∈ 푈푘 for any 푘 ≥ 2 · 푑(푥0, 푇푔) + |푔|. Thus, ⋃︀ 푈 = 퐺. Lastly, we have already seen that {id } ⊂ ⋂︀ 푈 . Now suppose 푔 ∈ 푈 푛∈ℤ 푛 푔 푛∈ℤ 푛 푛 for all 푛 ∈ ℤ. Then 푔 must be id퐺 since

∞ ⋃︁ 퐵푛(푥0) = 푋; 푛=0

that is, any such 푔 must necessarily ﬁx all of 푋, and therefore 푔 = id퐺 .

We will use this collection {푈푛}푛∈ℤ to deﬁne a function 푈 : (0, ∞) → Uid퐺 as in Lemma 5.4.6. But ﬁrst, let us make a few more observations regarding this collection of open neighborhoods. We

note that {푈푛}푛∈ℤ forms a countable base at id퐺:

+ Lemma 5.5.2. Let 푊 be any open neighborhood of id퐺 in 퐺. Then there exists an 푛 ∈ ℤ such

that 푈−푛 ⊂ 푊.

Proof. Since 푊 is open and contains id퐺, there exists some ﬁnite collection of sub-basis elements

풮(푓푖, 푇푖), 푖 = 1, . . . , 푘, each containing id퐺 such that

푘 ⋂︁ 풮(푓푖, 푇푖) ⊂ 푊. 푖=1 180 ⋃︀푘 Because id퐺 ∈ 풮(푓푖, 푇푖) for each 푖 = 1, . . . , 푘, we have 풮(푓푖, 푇푖) = 풮(id퐺, 푇푖). Let 풯 = 푖=1 푇푖

denote the minimal subtree in 푋 containing 푇푖 for each 푖 (one can construct 풯 by joining any disjoint 푇푖’s by geodesics in 푋). Because 풯 contains each 푇푖, we clearly have that 풮(id퐺, 풯 ) ⊂ ⋂︀푘 ⋂︀푘 푖=1 풮(id퐺, 푇푖). Now let 푔 ∈ 푖=1 풮(id퐺, 푇푖). So 푔(푥) = 푥 for all 푥 ∈ 푇푖 for every 푖 = 1, . . . , 푘.

Because 푔 is a graph automorphism, 푔 will ﬁx any geodesic joining the subtrees 푇푖. Hence, 푔 will ⋂︀푛 ﬁx any vertex in 풯 , and we have 푖=1 풮(id퐺, 푇푖) = 풮(id퐺, 풯 ).

Now we can choose a positive integer 푛 such that 풯 ⊂ 퐵푛(푥0). Then

풮(id퐺, 풯 ) ⊃ 푈−푛 = FixG(퐵푛(푥0))

and we have 푈−푛 ⊂ 푊 .

Now we show some product properties of elements in {푈푛}푛∈ℤ.

Lemma 5.5.3. Let 푛 ≤ 0.

1. 푈푛 is a subgroup of 퐺.

2. For any 푚 ∈ ℤ≤0

푈푛푈푚 = 푈푚푈푛 ⊂ 푈max(푛,푚).

3. For 푘 ∈ ℤ+,

푈푛푈푘, 푈푘푈푛 ⊂ 푈푘 = 푈max(푛,푘)

Proof. As we noted earlier, by deﬁnition 푈푛 for 푛 ≤ 0 is a point-wise stabilizer in 퐺, and hence is a subgroup. Hence (1) is true.

Now let 푛 ≤ 0 and 푚 ∈ ℤ≤0. If 푛 = 푚, then clearly 푈푛푈푚 = 푈푚푈푛 = 푈푛. So assume

(without loss of generality) that 푛 ≤ 푚. Let 푔 ∈ 푈푛 and 푕 ∈ 푈푚. Because 푛 ≤ 푚, we have

퐵|푚|(푥0) ⊂ 퐵|푛|(푥0). 181

If 푥 is any vertex in 퐵|푚|(푥0), then

푔(푕(푥)) = 푔(푥) = 푥,

푕(푔(푥)) = 푕(푥) = 푥.

If 푦 is any vertex in 퐵|푛|(푥0) ∖ 퐵|푚|(푥0), then

푔(푕(푦)) = 푕(푦),

푕(푔(푦)) = 푕(푦).

Thus, 푔푕 ∈ 푈푚푈푛 and 푕푔 ∈ 푈푚푈푛. Therefore 푈푛푈푚 = 푈푚푈푛 ⊂ 푈푚, and 푚 = max(푛, 푚). So (2) is proved. Next, note that because automorphisms preserve distance in 푋, we have for any 푔, 푕 ∈ 퐺,

푑(푥0, 푔푕(푥0)) ≤ 푑(푥0, 푔(푥0)) + 푑(푔(푥0), 푔푕(푥0))

= 푑(푥0, 푔(푥0)) + 푑(푥0, 푕(푥0))

+ Now suppose 푘 ∈ ℤ , and let 푔 ∈ 푈푛, 푕 ∈ 푈푘. Then 푑(푥0, 푔푕(푥0)) ≤ 0 + 푘 = 푘 and

푑(푥0, 푕푔(푥0)) ≤ 푘 + 0 = 푘. This proves (3).

Lemma 5.5.4. Let 푛 ≥ 1.

1. If 푈푛 contains a hyperbolic automorphism, then it is not a subgroup.

+ 2. For any 푚 ∈ ℤ , 푈푛푈푚 and 푈푚푈푛 are contained in 푈푛+푚.

Proof. Consider 푔 ∈ 푈푛 such that |푔| = 푛. This means 푥0 lies on the axis of 푔. Then:

2 푑(푔 (푥0), 푥0) = 2푛 182 2 2 2 and moreover, |푔 | = 2푛. So 푔 ∈ 푈2푛, but 푔 ∈/ 푈푛. So 푈푛 is not closed under multiplication, and hence cannot be a subgroup, and (1) is proved. Recall from the proof above that:

푑(푥0, 푔푕(푥0)) ≤ 푑(푥0, 푔(푥0)) + 푑(푔(푥0), 푔푕(푥0))

= 푑(푥0, 푔(푥0)) + 푑(푥0, 푕(푥0))

for any 푔, 푕 ∈ 퐺. If 푔 ∈ 푈푛 and 푕 ∈ 푈푚 with 푚 ≥ 1, then 푑(푥0, 푔푕(푥0)) ≤ 푛 + 푚, hence

푔푕 ∈ 푈푛+푚. Similarly, 푑(푥0, 푕푔(푥0)) ≤ 푚 + 푛 and so 푕푔 ∈ 푈푛+푚.

Now we can deﬁne a function 푈 : (0, ∞) → Uid퐺 as in Lemma 5.4.6. For each 푟 ∈ (0, ∞), we send 푟 ↦→ 푈(푟), where

⎧ ⎪ ⎪푈 푟 if 푟 ∈ (0, 1) ⎨ ⌈log2( )⌉ 푈(푟) = 1−푟 ⎪ ⎩⎪퐺 if 푟 ≥ 1

푟 For convenience, let 푓(푟) = so that we may write log (푓(푟)) = log (︀ 푟 )︀ . If 0 < 푟 ≤ 1 , 1 − 푟 2 2 1−푟 2 1 then log2(푓(푟)) ≤ 0, so 푈(푟) = 푈푚 where 푚 ≤ 0. If 2 < 푟 < 1, then log2(푓(푟)) > 0 and

푈(푟) = 푈푘 where 푘 ≥ 1. 2푚 Deﬁne 휌 : → (0, 1) by 휌(푚) = . Then 푈(휌(푚)) = 푈 for all 푚 ∈ , and moreover, ℤ 1 + 2푚 푚 ℤ

for all 푟 ∈ (0, 1), we have 푈(푟) = 푈푚 for some 푚 satisfying 휌(푚 − 1) < 푟 ≤ 휌(푚) (since 휌 is monotone). Now we must show that the function 푈(·) satisﬁes Lemma 5.4.6. We shall restate those conditions below.

Lemma 5.5.5. The function 푈 : (0, ∞) → Uid퐺 deﬁned above satisﬁes the following conditions:

1. For all 푟 > 1, 푈(푟) = 퐺;

2. For all 0 < 푟 < 푠, 푈(푟) ⊂ 푈(푠); 183 ⋃︀ 3. For all 푠 > 0, 푟<푠 푈(푟) = 푈(푠);

1 4. For every neighborhood 푁 of 푒, there is an 푛 ∈ ℕ such that 푈( 푛 ) ⊂ 푁;

5. For each 푛 ∈ ℕ, there exists a neighborhood 푁 of id퐺 such that

(︂ 1 )︂ 푈(푟)푁 ⊂ 푈 푟 + 푛

holds for all 푟 > 0.

Proof. By deﬁnition, (1) holds. Note that, if 푟 ∈ (0, 1) and 푠 ≥ 1, then we trivially have that

푈(푟) ⊂ 푈(푠). Now suppose 푟, 푠 ∈ (0, 1) with 푟 < 푠. Then since log2(푓(푟)) is a strictly increasing function, we have

log2(푓(푟)) < log2(푓(푠))

⇒ ⌈log2(푓(푟))⌉ ≤ ⌈log2(푓(푠))⌉

⇒ 푈(푟) ⊂ 푈(푠)

since the collection {푈푛}푛∈ℤ forms an ascending chain via inclusion. This proves (2). Now we show (3). First, suppose 푠 ≥ 1. By (2), if 0 < 푟 < 푠, then 푈(푟) ⊂ 푈(푠) = 퐺. So we clearly have ⋃︁ 푈(푟) ⊂ 푈(푠) = 퐺. 푟<푠

But as 푟 runs through (0, 1), 푈(푟) yields 푈푛 for all 푛 ∈ ℤ. By Lemma 5.5.1, we know that ⋃︀ 푈 = 퐺, thus 푛∈ℤ 푛 ⋃︁ 푈(푟) = 퐺 = 푈(푠). 푟<푠 ⋃︀ Now suppose 0 < 푠 < 1. Again, by (2), we have 푟<푠 푈(푟) ⊂ 푈(푠). And clearly, for any 푦 ∈ ℝ, we know that there exists 푥 < 푦 ∈ ℝ such that ⌈푥⌉ = ⌈푦⌉. Thus, there exists an 푟′ ∈ (0, 1) with 184 ′ ′ ′ 푟 < 푠 such that ⌈log2(푓(푟 ))⌉ = ⌈log2(푓(푠))⌉, meaning 푈(푟 ) = 푈(푠). Hence,

⋃︁ 푈(푠) ⊂ 푈(푟) 푟<푠

and (3) is proved.

Now let 푊 be an open neighborhood of id퐺. From Lemma 5.5.2, we have for some 푚 ≤ 0,

1 푈푚 ⊂ 푊. So to prove (4), we just need to solve for 푟 = 푛 in the equation:

⌈︂ (︂ 1 )︂⌉︂ ⌈︂ (︂ )︂⌉︂ 푛 1 log2 1 = log2 = 푚 1 − 푛 푛 − 1

Such an 푛 can certainly be chosen so that

(︂ 1 )︂ 푚 − 1 < log ≤ 푚 2 푛 − 1

(︀ 1 )︀ by continuity. Thus, we have an 푛 ∈ ℕ with 푈 푛 ⊂ 푊 , proving (4). Now let 푛 ∈ ℕ. If 푛 = 1, then 푈(푟 + 1) = 퐺 for all 푟 > 0. So in this case, let 푁 = 퐺 and we are done. So assume 푛 ≥ 2, and let 푟 > 0. If 푟 = 1, we can also let 푁 = 퐺 and be done, since

(︀ 1 )︀ here 푈 푟 + 푛 = 퐺. So now assume that 0 < 푟 < 1. We have 푈(푟) = 푈푘 for some 푘 ∈ ℤ. Now set ⌈︂ (︂ 1 )︂⌉︂ 푚 = log 2 푛 − 1

1 and note that 푚 ≤ 0 since 푛 ≥ 2. Then by (4), 푈( 푛 ) = 푈푚 = 푁 is an open neighborhood of id푔, and

푈(푟)푁 = 푈푘푈푚 = 푈max(푘,푚)

1 1 1 by Section 5.5. Because 푟 < 푟 + 푛 and 푛 < 푟 + 푛 , no matter which of 푘 and 푚 is larger, we’ll have (︂ 1 )︂ 푈(푟)푁 ⊂ 푈 푟 + 푛

by (2). This completes the proof. 185 Now we can deﬁne the function 휙 as in Lemma 5.4.6,

휙(푔) = inf {푟 ∈ (0, 1] | 푔 ∈ 푈(푟)} , and then from Lemma 5.4.5, we can deﬁne a left-invariant metric on 퐺:

휎(푔, 푕) = sup{|휙(푧푔) − 휙(푧푕)| : 푧 ∈ 퐺} for 푔, 푕 ∈ 퐺. We also have a norm ‖·‖ : 퐺 → ℝ+. From Lemma 5.4.5, we recall that the norm function and the 휙 function are related as follows:

‖푔‖ = 푑(푔, id퐺),

휙(푔) = min{‖푔‖, 1}.

Additionally, from Lemma 5.4.6, we have

푈(푟) = {푔 ∈ 퐺 | 휙(푔) < 푟}.

We now record some computational facts concerning our metric.

Lemma 5.5.6. For all 푔 ∈ 퐺, 휎(푔, id퐺) = 휙(푔)

Proof. This follows immediately from our construction and the remarks above. We know that 0 ≤ 휙(푔) ≤ 1 for all 푔 ∈ 퐺. And

휙(푔) = min{‖푔‖, 1} = min{휎(푔, id퐺), 1} 186 But for a ﬁxed 푔 ∈ 퐺, we have

휎(푔, id퐺) = sup |휙(푧푔) − 휙(푧)| ≤ 1. 푧∈퐺

Thus, 휙(푔) = 휎(푔, id퐺).

Proposition 5.5.7. The ball of radius 휀 > 0 centered at id퐺 in 퐺 is given by

⎧ ⎪ ⎨⎪푈(휌(푚)) = 푈푚 if 휌(푚 − 1) < 휀 ≤ 휌(푚) 퐵휎(id퐺, 휀) = ⎪ ⎩⎪퐺 if 휀 ≥ 1

Proof. This is clear if 휀 ≥ 1. Indeed, the inclusion 퐵휎(id퐺, 휀) ⊂ 퐺 is by deﬁnition. For the other, let 푔 ∈ 퐺. Since 휎(푔, id퐺) = 휙(푔) ≤ 1 always holds, we have 푔 ∈ 퐵휎(id퐺, 휀).

Now assume 0 < 휀 < 1. Letting 푔 ∈ 퐵휎(id퐺, 휀), then 휎(푔, id퐺) = 휙(푔) < 휀. Because

휙(푔) < 휀 implies 푈(휙(푔)) ⊂ 푈(휀), we have 푔 ∈ 푈(휀) = 푈푚, where 푚 is an integer satisfying

휌(푚 − 1) < 휀 ≤ 휌(푚). Thus 퐵휎(id퐺, 휀) ⊂ 푈푚.

Now suppose 푔 ∈ 푈푚 where 휌(푚 − 1) < 휀 ≤ 휌(푚). Then 푔 ∈ 푈(휀), and we can choose a positive integer 푁 = 푁(휀) such that

1 휌(푚 − 1) < 휀 − < 휌(푚) 푁

1 Hence 푔 ∈ 푈(휀− 푁 ), and we can conclude that 휎(푔, id퐺) = 휙(푔) < 휀. Thus 푈푚 ⊂ 퐵휎(id퐺, 휀).

This last result will allow us to simplify the computation of the Chabauty metric, which we carry out in the next section.

Lemma 5.5.8. Let 휀 > 0 and 퐹 ⊂ 퐺 a closed subset of 퐺. Then 풱휀(퐹 ) = 퐹 푈(휀), where 풱휀(퐹 ) is deﬁned as

풱휀(퐹 ) = {푔 ∈ 퐺 | 휎(푔, 퐹 ) < 휀}, the 휀-neighborhood of 퐹 in 퐺. 187

Proof. Firstly, suppose 휀 ≥ 1. Then 풱휀(퐹 ) = 퐺 by the above proposition, and clearly 퐺 = 퐹 퐺 = 푈(휀).

Now suppose 0 < 휀 < 1 and let 푔 ∈ 풱휀(퐹 ). Recall that

휎(푔, 퐹 ) = inf{휎(푔, 푕) | 푕 ∈ 퐹 }.

Hence, there exists a sequence (푕푛)푛≥1 in 퐹 such that 휎(푔, 푕푛) converges to 휎(푔, 퐹 ) < 휀. Thus we can choose a positive integer 푁 such that

|휎(푔, 푕푁 ) − 휎(푔, 퐹 )| < 휀 − 휎(푔, 퐹 ).

It follows that 휎(푔, 푕푁 ) ≤ 휎(푔, 퐹 ) + |휎(푔, 푕푁 ) − 휎(푔, 퐹 )| < 휀. Then by the left-invariance of the metric 휎,

−1 휎(푔, 푕푁 ) = 휎(푕푁 푔, id퐺) < 휀.

−1 So by the above proposition, 푕푁 푔 ∈ 푈(휀) = 푈푚, where 푚 ∈ ℤ such that 휌(푚−1) < 휀 ≤ 휌(푚) < −1 1. Thus, 푔 = 푕푁 (푕푁 푔) ∈ 퐹 푈(휀) and 풱휀(퐹 ) ⊂ 퐹 푈(휀). Now let 푕 ∈ 퐹 and 푔 ∈ 푈(휀). Then by left-invariance of the metric,

휎(푕푔, 푕) = 휎(푔, id퐺) < 휀

since 푔 ∈ 푈(휀). Then it is clear that 휎(푕푔, 퐹 ) < 휀 and hence 푕푔 ∈ 풱휀(퐹 ). Therefore, 퐹 푈(휀) ⊂

풱휀(퐹 ).

5.6 The Chabauty Space for a Twin Tree

Now we will combine all of our above results to discuss the Chabauty space for a twin tree 푇 = (푇 +, 푇 −, 훿*). Let 퐺 ≤ 퐴0 ≤ 퐴 = Aut(푇 ) and suppose that 퐺 acts strongly transitively on 푇 . We will assume that 푇 is thick, i.e. the valency of every vertex in either tree is greater than

0 −1 or equal to 3. The example to keep in mind here is that of GL2(푘[푡, 푡 ]) acting on the twinning 188

described in Chapter 1 (which come from twinning the Bruhat-Tits trees associated to GL2(푘[푡])

−1 and GL2(푘[푡 ])).

+ − + + Fix a twin apartment Σ = (Σ , Σ ). We denote the vertices of Σ by 푥푛 , 푛 ∈ ℤ and likewise

− − + − the vertices of Σ by 푥푛 , so that opposite vertices match identically by subscript: 푥푖 opp 푥푖 . Note that 퐺 can be given a topology using either tree 푇 + or 푇 −. That is, for 휀 ∈ {+, −}, we

휀 can topologize 퐺 using sub-basis elements 풮 (푓, 푋) = {푔 ∈ 퐺 | 푓|푋 = 푔|푋 }, where 푓 ∈ 퐺 and 푋 ⊂ 푇 휀 is a ﬁnite subtree.

− − − We choose to topologize 퐺 using 푇 . For the topology, we ﬁx the point 푥0 ∈ 푇 . Using the results from Section 5.5, we know that 퐺 is metrizable, and we have a left-invariant metric on 퐺, which we denote by 휎. Now recall that the space of closed subsets ℱ(퐺) inherits a metric, denoted

푑Hau. Then for any 퐹1, 퐹2 ∈ ℱ(퐺),

⎧ ⃒ ⎫ ⃒ 푐 푐 ⎪ ⃒ 퐹1 ∪ 퐵휎(id퐺, 1/휀) ⊂ 풱휀(퐹2 ∪ 퐵휎(id퐺, 1/휀)) ⎪ ⎪ ⃒ ⎪ ⎨ ⃒ ⎬ 푑Hau(퐹1, 퐹2) = inf 휀 > 0 ⃒ ⃒ ⎪ ⃒ ⎪ ⎪ ⃒ 푐 푐 ⎪ ⎩ ⃒ 퐹2 ∪ 퐵휎(id퐺, 1/휀) ⊂ 풱휀(퐹1 ∪ 퐵휎(id퐺, 1/휀)) ⎭

Recall that we proved in Section 5.5 that balls centered at id퐺 via 휎 are given by

⎧ ⎪ ⎨⎪푈(휌(푚)) = 푈푚 if 휌(푚 − 1) < 휀 ≤ 휌(푚) 퐵휎(id퐺, 휀) = ⎪ ⎩⎪퐺 if 휀 ≥ 1

And for 퐹 ∈ ℱ(퐺), the 휀-neighborhood of 퐹 is given by:

풱휀(퐹 ) = 퐹 푈(휀)

Hence, we have the following simpliﬁcation to the Chabauty metric 푑Hau: 189

Lemma 5.6.1. For all 퐹1, 퐹2 ∈ ℱ(퐺),

⎧ ⃒ ⎫ ⃒ ⎨⎪ ⃒ 퐹1 ⊂ 풱휀(퐹2) ⎬⎪ ⃒ 푑Hau(퐹1, 퐹2) = inf 휀 > 0 ⃒ ⎪ ⃒ 퐹 ⊂ 풱 (퐹 ) ⎪ ⎩ ⃒ 2 휀 1 ⎭

푐 Proof. When 휀 ≤ 1, we have 퐵휎(id퐺, 1/휀) = 퐺. Thus, 퐵휎(id퐺, 1/휀) = ∅ and so we can drop the balls from the computation.

Distances Between Point Stabilizers in Chabauty Metric

In this section, we want to measure the distance between two point stabilizers, where the points lie in the apartment Σ+ (recall that we topologized 퐺 using the 푇 − tree). So we want to compute

+ + 푑Hau(퐺 + , 퐺 + ) for some integers 푖 and 푗, where 퐺 + = Stab퐺(푥 ) and 퐺 + = Stab퐺(푥 ). From 푥푖 푥푗 푥푖 푖 푥푗 푗 the lemma above, this means we must compute the following inﬁmum:

⎧ ⃒ ⎫ ⎪ ⃒ ⎪ ⎨ ⃒ 퐺푥+ ⊂ 퐺푥+ 푈(휀) ⎬ ⃒ 푖 푗 inf 휀 > 0 ⃒ ⎪ ⃒ 퐺 + ⊂ 퐺 + 푈(휀) ⎪ ⎩ ⃒ 푥푗 푥푖 ⎭

First, we need the following lemmas, the ﬁrst of which concerns the existence of opposite paths.

휀 휀 휀 휀 휀 휀 Lemma 5.6.2. Let 휀 ∈ {+, −}. Let 푙 = (푐0, 푐1, . . . , 푐푘−1) be a reduced path in 푇 , with 푐푖 =

휀 휀 −휀 휀 −휀 −휀 −휀 −휀 [푣푖 , 푣푖+1]. Suppose 푦 opp 푣푘 with 푦 ∈ 푉 푇 . Then there exists a reduced path 푙 in 푇 ,

−휀 −휀 −휀 −휀 −휀 −휀 −휀 −휀 −휀 −휀 휀 푙 = (푑0 , 푑1 , . . . , 푑푘−1) with 푑푖 = [푥푖 , 푥푖+1] and 푥푘 = 푦 and such that 푑푖 opp 푐푖 for 푖 = 0, 1, 2, . . . , 푘 − 1.

Proof. Without loss of generality, let 휀 = + so that −휀 = −. We proceed by induction on 푘, the

+ + + length of the reduced path 푙 = (푐0 , . . . , 푐푘−1).

+ + + Base Case: Let 푘 = 1. Then 푙 consists of a single edge 푐0 with initial vertex 푣0 and terminal

+ − + * + − * + − vertex 푣1 . Let 푦 be any vertex opposite 푣1 . By deﬁnition, 훿 (푣1 , 푦 ) = 0 and thus 훿 (푣0 , 푦 ) =

− − * + − 1. Hence, there exists a unique vertex 푧 adjacent to 푦 such that 훿 (푣0 , 푧 ) = 2. 190 Since the twinned trees 푇 + and 푇 − are thick, we have that the valency of 푦− is at least 3.

− − − Hence, we have that there exists at least two more neighbors of 푦 , say 푥1 and 푥2 . Note that the

+ codistance for these vertices with 푣1 is 1:

* + − * + − 훿 (푣1 , 푥1 ) = 1, 훿 (푣1 , 푥2 ) = 1

− − + − Since 푧 is the unique vertex adjacent to 푦 having a codistance of 2 with 푣0 , we must have that 푥1

− − − and 푥2 (and any other adjacent vertex) have codistance 0. Hence, we have a path 푙 = (푑0 ) with

− − − + − + − + + + 푑0 = [푥1 , 푦 ], 푣0 opp 푥1 and 푣1 opp 푦 . In other words, we have opposite edges 푐0 = [푣0 , 푣1 ]

− − − and 푑0 = [푥1 , 푦 ]. Inductive Step: Assume that 푘 ≥ 1 and that the hypothesis holds for any path in 푇 + of this length. We want to show that the same is true for paths of length 푘 + 1.

+ + + + + + + + − Let 푙 = (푐0 , 푐1 , . . . , 푐푘 ), 푐푖 = [푣푖 , 푣푖+1], be a reduced path in 푇 and let 푦 be a vertex

+ + + + + + opposite 휔(푐푘 ) = 푣푘+1. Consider the sub-path ̃︀푙 = (푐1 , 푐2 , . . . , 푐푘 ). By the inductive hypothesis,

′ − − − − − − − − + − there exists a reduced path 푙 = (푑1 , 푑2 , . . . , 푑푘 ) with 푑푖 = [푥푖 , 푥푖+1], 휔(푑푘 ) = 푦 and 푐푖 opp 푑푖 for 푖 = 1, 2, . . . , 푘.

+ − * + − Because 푐1 is opposite to 푑1 , we have 훿 (푣1 , 푥1 ) = 0. So now, by a similar argument as in the

* − + − − base case, we have that 훿 (푥1 , 푣0 ) = 1; hence there exists a unique vertex 푧 adjacent to 푥1 such

* − + − − that 훿 (푧 , 푣0 ) = 2. Then by thickness of 푇 , we have that 푥1 has at least one more neighboring

− − − − + vertex, say 푥0 ̸= 푥2 . By uniqueness of 푧 , 푥0 must be opposite to 푣0 . Hence, we can tack on the

− − − ′ − − − − − edge 푑0 = [푥0 , 푥1 ] to the path 푙 and get the path 푙 = (푑0 , 푑1 , . . . , 푑푘 ), which has each edge 푑푗 + opposite to 푐푗 , as desired.

Our second lemma gives us a relation between the distance function 푑휀 and the co-distance function.

Lemma 5.6.3. Let 푥, 푦 be vertices in 푇 휀 and 푧 a vertex of 푇 −휀. Then,

|푑휀(푥, 푦) − 훿*(푥, 푧)| ≤ 훿*(푦, 푧) ≤ 푑휀(푥, 푦) + 훿*(푥, 푧). 191 휀 휀 휀 휀 휀 휀 Proof. Let 푙 = (푐0, 푐1, . . . , 푐푛−1) be the unique reduced path in 푇 from 푥 to 푦. Then 푑 (푥, 푦) = 푛 = |푙휀|. Set 푚 = 훿*(푥, 푧). Recall that 푚 is the minimal distance in 푇 −휀 from the vertex 푧 to a vertex 푥op opposite to 푥. Hence, there is a minimal reduced path 푝 from 푥op to 푧 such that 푑−휀(푥op, 푧) = 푚 = |푝|.

−휀 −휀 −휀 op By the previous Lemma 5.6.2, there exists a minimal path 푙 = (푑0 , . . . , 푑푛−1) starting at 푥

op 휀 −휀 and ending at a vertex 푦 that is opposite to 푦, and such that 푐푖 opp 푑푖 for all 푖 = 0, 1, . . . , 푛 − 1. Then necessarily,

훿*(푦, 푧) ≤ |푙−휀| + |푝| = 푛 + 푚 = 푑휀(푥, 푦) + 훿*(푥, 푧).

Moreover, the smallest possible length of a reduced path from 푧 to 푦op is |푛 − 푚|, hence the lower bound.

+ Now let 푥푖, 푥푗 ∈ 푉 Σ . Set 퐺푖 = 퐺 + and 퐺푗 = 퐺 + . To compute 푑Hau(퐺푖, 퐺푗) we must show 푥푖 푥푗 that:

1. 퐺푖 ⊆ 퐺푗푈(휀),

2. 퐺푗 ⊆ 퐺푖푈(휀), and we must ﬁnd the smallest 휀 > 0 for which these inclusions hold. Recall that for 푚 ∈ ℤ, we have 푈(휀) = 푈푚 if and only if 휌(푚 − 1) < 휀 ≤ 휌(푚). Our theorem below gives the best possible values of 푚.

Theorem 5.6.4. We have the following inclusions:

1. If 0 ≤ |푗| < |푖|, then for 푚 = 2|푖 − 푗|

퐺푗 ⊂ 퐺푖푈푚

퐺푖 ⊂ 퐺푗푈푚

Moreover, 퐺푖 ⊈ 퐺푗푈푘 for any 푘 < 푚. 192 2. If 푗 ≤ 0 < 푖, then for 푚 = max(2|푖|, 2|푗|)

퐺푗 ⊂ 퐺푖푈푚

퐺푖 ⊂ 퐺푗푈푚

Moreover, 퐺푖 ⊈ 퐺푗푈푘 for any 푘 < 푚.

Proof. First we show (1). Suppose we have 0 ≤ 푗 < 푖. Let 푔 ∈ 퐺푖. Then based on our labelling of the twin apartment Σ,

* + − * + − * + − 푖 = 훿 (푥푖 , 푥0 ) = 훿 (푔푥푖 , 푔푥0 ) = 훿 (푥푖 , 푔푥0 ).

We want to show that 푔 ∈ 퐺푗푈푚, for some integer 푚. To that end, we must compute bounds on

+ − the co-distance between 푥푗 and the moved vertex 푔푥0 . By Lemma 5.6.3 we see that

+ + + * + − * + − + + + * + − |푑 (푥푖 , 푥푗 ) − 훿 (푥푖 , 푔푥0 )| ≤ 훿 (푥푗 , 푔푥0 ) ≤ 푑 (푥푖 , 푥푗 ) + 훿 (푥푖 , 푔푥0 )

* + − |(푖 − 푗) − 푖| ≤ 훿 (푥푗 , 푔푥0 ) ≤ (푖 − 푗) + 푖

* + − 푗 ≤ 훿 (푥푗 , 푔푥0 ) ≤ 2푖 − 푗.

* + − Now set 푁 = 훿 (푥푗 , 푔푥0 ). By strong transitivity of 퐺, an 푕 ∈ 퐺푗 can be chosen such that

+ − + − − − * + − the pair (푥푗 , 푔푥0 ) is sent to the pair (푥푗 , 푥푙 ) where 푥푙 is a vertex on Σ and 훿 (푥푗 , 푥푙 ) = 푁.

− − − − − There are at most two such vertices on Σ : 푥푙 = 푥푗+푁 or 푥푙 = 푥푗−푁 . Since 푗 ≤ 푁 ≤ 2푖 − 푗, − − − we have two possible extremes: the vertex 푥푗−푁 lies between the vertices 푥2푗−2푖 and 푥0 ; and the − − − − − vertex 푥푗+푁 lies between 푥2푗 and 푥2푖. Now we compute the possible distances between 푥푙 and 푥0 :

− − − 0 ≤ 푑 (푥0 , 푥푗−푁 ) ≤ |2푗 − 2푖| = 2(푖 − 푗),

− − − 2푗 ≤ 푑 (푥0 , 푥푗+푁 ) ≤ 2푖. 193 − − Hence, 푕 ∈ 퐺푖 can be chosen such that 푕푔(푥0 ) = 푥푙 satisﬁes

− − − 푑 (푕푔(푥0 ), 푥0 ) ≤ 2(푖 − 푗).

−1 Then 푔 = 푕 (푕푔) ∈ 퐺푗푈푚 with 푚 = 2(푖 − 푗).

Now we show show the second inclusion, 퐺푗 ⊆ 퐺푖푈푚, for 0 ≤ 푗 < 푖. Let 푔 ∈ 퐺푗. Then

* + − * + − * + − 푗 = 훿 (푥푗 , 푥0 ) = 훿 (푔푥푗 , 푔푥0 ) = 훿 (푥푗 , 푔푥0 ).

Applying Lemma 5.6.3 it follows that:

+ + + * + − * + − + + + * + − |푑 (푥푗 , 푥푖 ) − 훿 (푥푗 , 푔푥0 )| ≤ 훿 (푥푖 , 푔푥0 ) ≤ 푑 (푥푗 , 푥푖 ) + 훿 (푥푗 , 푔푥0 )

* + − |(푖 − 푗) − 푗| ≤ 훿 (푥푖 , 푔푥0 ) ≤ 푖 − 푗 + 푗

* + − |푖 − 2푗| ≤ 훿 (푥푖 , 푔푥0 ) ≤ 푖.

* + − Set 푀 = 훿 (푥푖 , 푔푥0 ). Since 퐺 acts strongly transitively on the twin tree, we can choose an 푕 ∈ 퐺푖

+ − + − − − − − − sending the pair (푥푖 , 푔푥0 ) to the pair (푥푖 , 푥푘 ), where 푥푘 is a vertex on Σ with 푑 (푥푖 , 푥푘 ) = 푀.

− − − − − There are at most two such vertices on Σ : 푥푘 = 푥푖+푀 or 푥푘 = 푥푖−푀 . Since |푖 − 2푗| ≤ 푀 ≤ 푖, the possibilities are a little different from those in the ﬁrst case. Note that, if 0 ≤ 푗 < 2푗 < 푖, then

− 2푗 < 2(푖 − 푗) = |2푗 − 2푖| < 2푖 and if 0 ≤ 푗 < 푖 ≤ 2푗, then 2(푖 − 푗) ≤ 2푗. Then the vertex 푥푖−푀 − − − − lies between the vertices 푥0 and 푥min(2푗,2(푖−푗)); and the vertex 푥푖+푀 lies between 푥max(2푗,2(푖−푗)) and − 푥2푖. Hence, if 2푗 < 푖 then

− − − 0 ≤ 푑 (푥0 , 푥푖−푀 ) ≤ 2푗 < 2(푖 − 푗),

− − − 2(푖 − 푗) ≤ 푑 (푥0 , 푥푖+푀 ) ≤ 2푖. 194 Else, if 푗 < 푖 ≤ 2푗, then

− − − 0 ≤ 푑 (푥0 , 푥푖−푀 ) ≤ 2(푖 − 푗),

− − − 2푗 ≤ 푑 (푥0 , 푥푖+푀 ) ≤ 2푖.

− − So regardless of the case, an element 푕 ∈ 퐺푖 can be chosen such that 푕푔(푥0 ) = 푥푙 satisﬁes

− − − 푑 (푕푔(푥0 ), 푥0 ) ≤ 2(푖 − 푗).

Thus we have 푔 ∈ 퐺푖푈푚, where 푚 = 2(푖 − 푗). Lastly, we show that 푚 = 2(푖 − 푗) is the best possible outcome for 0 ≤ 푗 < 푖. In particular, we

+ − have that 퐺푖 ⊈ 퐺푗푈푘 for 푘 < 2(푖 − 푗). Consider the automorphism 푠푖 that ﬁxes 푥푖 and 푥푖 , and − − + − swaps the vertices 푥푖−1 and 푥푖+1. In other words, 푠 is a reﬂection through the vertices 푥푖 and 푥푖 in the twin apartment Σ. By an equivalence of 퐺 acting strongly transitively on 푇 , we have that 푠 ∈ 퐺 ([AB08, 6.3, Lemma 6.70]).

− − − − − Note that 푠푖(푥0 ) = 푥2푖 and so 푑 (푠푖푥0 , 푥0 ) = 2푖. Consider any 푕 ∈ 퐺푗. Then

* + − * + − * + − 푗 = 훿 (푥푗 , 푥0 ) = 훿 (푕푥푗 , 푕푥0 ) = 훿 (푥푗 , 푕푥0 ).

By Lemma 5.6.3,

* + − − − − * + − 훿 (푥푗 , 푠푖푥0 ) ≤ 푑 (푕푥0 , 푠푖푥0 ) + 훿 (푥푗 , 푕푥0 )

− − − = 푑 (푕푥0 , 푥2푖) + 푗.

Hence,

− − − * + − 푑 (푕푥0 , 푥2푖) ≥ 훿 (푥푗 , 푥2푖) − 푗 = (2푖 − 푗) − 푗 = 2(푖 − 푗).

− − − − − −1 − Since 푑 (푕푥0 , 푠푖푥0 ) = 푑 (푥0 , 푕 푠푖(푥0 )), we have that if 푠푖 = 푔푢 for 푔 ∈ 퐺푗 and 푢 ∈ 푈푘, then we must have 푘 ≥ 2(푖 − 푗). This proves the inclusions in (1) are the best possible simultaneous 195 inclusions for the case 0 ≤ 푗 < 푖.

To ﬁnish (1), we must show that 퐺−푖 ⊆ 퐺−푗푈푚 and 퐺−푗 ⊆ 퐺−푖푈푚, with 푚 = 2|푖 − 푗|. But this follows from what we have just shown, since, for any integer 푛,

푠0퐺푛푠0 = 퐺−푛

+ − where 퐺 = 퐺 + and 푠 is the reﬂection in Σ across 푥 and 푥 . Hence, 푛 푥푛 0 0 0

퐺−푖 = 푠0퐺푖푠0 ⊆ 푠0퐺푗푈푚푠0

= 푠0퐺푗푠0푠0푈푚푠0

= 퐺−푗푈푚,

where 푠0푈푚푠0 = 푈푚 since 푠0 ∈ 퐺 − . It follows by the same argument that 퐺−푗 ⊆ 퐺−푖푈푚 and 푥0 that 퐺−푖 ⊈ 퐺−푗푈푘, for 푘 < 2|푖 − 푗|. This completely proves case (1). Now we show case (2). Suppose that 푗 ≤ 0 < 푖. Our argument is exactly the same as in (1).

* − + * − + Firstly, let 푔 ∈ 퐺푖. Then 푖 = 훿 (푥0 , 푥푖 ) = 훿 (푔푥0 , 푥푖 ). Then by Lemma 5.6.3,

+ + + * − + * + − + + + * − + |푑 (푥푖 , 푥푗 ) − 훿 (푔푥0 , 푥푖 )| ≤ 훿 (푥푗 , 푔푥0 ) ≤ 푑 (푥푖 , 푥푗 ) + 훿 (푔푥0 , 푥푖 )

* + − |푖 − 푗 − 푖| ≤ 훿 (푥푗 , 푔푥0 ) ≤ 푖 − 푗 + 푖

* + − −푗 ≤ 훿 (푥푗 , 푔푥0 ) ≤ 2푖 − 푗.

* + − + − + − − Now we set 푁 = 훿 (푥푗 , 푔푥0 ) and choose 푕 ∈ 퐺푗 sending (푥푗 , 푔푥0 ) to (푥푗 , 푥푙 ) where 푥푙 is the − − − − − vertex 푥푗−푁 or the vertex 푥푗+푁 . The vertex 푥푗−푁 lies between 푥2푗−2푖 and 푥2푗, and hence

− − − 2|푗| ≤ 푑 (푥0 , 푥푗−푁 ) ≤ 2|푖 − 푗|. 196 − − − The vertex 푥푗+푁 lies between 푥0 and 푥2푖 and hence

− − − 0 ≤ 푑 (푥0 , 푥푗+푁 ) ≤ 2푖.

− − − − Thus, by choosing 푕 ∈ 퐺푗 that sends 푔푥0 to 푥푗+푁 , we guarantee that 푑 (푥0 , 푕푔(푥0 )) ≤ 2푖.

−1 Hence, 푔 = 푕 (푕푔) ∈ 퐺푗푈2푖.

* + − * + − Now let 푔 ∈ 퐺푗. Then |푗| = 훿 (푥푗 , 푥0 ) = 훿 (푥푗 , 푔푥0 ) and by Lemma 5.6.3,

+ + + * − + * + − + + + * − + |푑 (푥푗 , 푥푖 ) − 훿 (푔푥0 , 푥푗 )| ≤ 훿 (푥푖 , 푔푥0 ) ≤ 푑 (푥푖 , 푥푗 ) + 훿 (푔푥0 , 푥푗 )

* + − |푖 − 푗 − |푗|| ≤ 훿 (푥푖 , 푔푥0 ) ≤ 푖 − 푗 + |푗|

* + − |푖 − 푗 + 푗| ≤ 훿 (푥푖 , 푔푥0 ) ≤ 푖 − 푗 − 푗

* + − 푖 ≤ 훿 (푥푖 , 푔푥0 ) ≤ 푖 − 2푗.

* + − − − − Set 푀 = 훿 (푥푖 , 푔푥0 ). The vertex 푥푖−푁 lies between the vertices 푥2푗 and 푥0 on the apartment Σ . Hence,

− − − 0 ≤ 푑 (푥0 , 푥푖−푁 ) ≤ 2|푗|.

− − Similarly, the vertex 푥푖+푁 lies between 푥2푖 and 푥2(푖−푗) and hence

− − − 2푖 ≤ 푑 (푥0 , 푥푖+푁 ) ≤ 2(푖 − 푗).

− − − − Thus, we can choose 푕 ∈ 퐺푖 such that 푕푔(푥0 ) = 푥푖−푁 and guarantee that 푑 (푥0 , 푕푔(푥0 )) ≤ 2|푗|.

−1 Hence, 푔 = 푕 (푕푔) ∈ 퐺푖푈2|푗|. Lastly, we show that the inclusion 퐺푖 ⊆ 퐺푗푈푚 is strict for

푗 ≤ 0 < 푖, where 푚 = max(2푖, 2|푗|). As in the case of (1), we let 푕 ∈ 퐺푗 and apply Lemma 5.6.3:

* + − − − − * − + 훿 (푥푗 , 푠푖푥0 ) ≤ 푑 (푕푥0 , 푠푖푥0 ) + 훿 (푕푥0 , 푥푗 )

− − − = 푑 (푕푥0 , 푥2푖) + |푗|. 197 Hence,

− − − * + − 푑 (푕푥0 , 푠푖푥0 ) ≥ 훿 (푥푗 , 푠푖푥0 ) − |푗| = |푗| + 2푖 − |푗| = 2푖.

Thus 푠푖 ∈/ 퐺푗푈푘 for 푘 < 2푖 ≤ 푚, which shows that the inclusions in (2) are the best possible. This completes the proof.

Using the previous theorem, it is a simple manner to compute 푑Hau(퐺푖, 퐺푗). Recall that the

function 휌 : ℤ → (0, 1) is given by 2푛 휌(푛) = . 1 + 2푛

Corollary 5.6.5. If 0 ≤ |푗| < |푖|, then

푑Hau(퐺 + , 퐺 + ) = 휌(2|푖 − 푗| − 1). 푥푖 푥푗

If 푗 ≤ 0 < 푖 and 푚 = max(2푖, 2|푗|), then

푑Hau(퐺 + , 퐺 + ) = 휌(푚 − 1). 푥푖 푥푗

Proof. Recall that for 휀 ∈ (0, 1), 푈(휀) = 푈푚 if and only if 휌(푚 − 1) < 휀 ≤ 휌(푚). If 0 ≤ |푗| < |푖|, then from Theorem 5.6.4 we know that the smallest possible value for 푚 is 2|푖 − 푗|. Hence,

푑Hau(퐺 + , 퐺 + ) = inf{휀 > 0 | 휌(2|푖 − 푗| − 1) < 휀 ≤ 휌(2|푖 − 푗|)} 푥푖 푥푗 = 휌(2|푖 − 푗| − 1).

Likewise, for 푗 ≤ 0 < 푖 we know the best possible inclusions occur for 푚 = max(2푖, 2|푗|). Hence,

푑Hau(퐺 + , 퐺 + ) = inf{휀 > 0 | 휌(푚 − 1) < 휀 ≤ 휌(푚)} 푥푖 푥푗 = 휌(푚 − 1). 198 + + + As an example, suppose we have adjacent vertices 푥푖 and 푥푖+1 in Σ . Then the distance in

풞(퐺) between 퐺 + and 퐺 + is: 푥푖 푥푖+1

21 2 푑Hau(퐺푥+ , 퐺푥+ ) = 휌(2 − 1) = = . 푖 푖+1 1 + 21 3

As a ﬁnal remark, it is interesting to see that the Chabauty distances for point stabilizers of

+ vertices is different depending on which side of the vertex 푥0 one lies. For instance, consider

+ + + + the pairs of vertices 푥1 and 푥4 , and 푥−1 and 푥2 . Their corresponding point stabilizers in 퐺 have distances:

32 푑 (퐺 , 퐺 ) = 휌(5) = , Hau 1 4 33 8 푑 (퐺 , 퐺 ) = 휌(3) = . Hau −1 2 9

And yet, each pair of vertices is the same distance apart in 푇 +.

5.7 Future Work

The above is preliminary work to better understanding the Chabauty space of a group acting on a twin tree. As we have seen, point stabilizers can be viewed as non-uniform lattices in the automorphism group of the twinning. One future goal would be to collect as much information as possible regarding uniform and non-uniform lattices in 풞(퐺) and try to better understand how they

ﬁt together using the metric 푑Hau on 풞(퐺). 199

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