Covering Spaces and the Fundamental Group
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An Ascending Hnn Extension
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 134, Number 11, November 2006, Pages 3131–3136 S 0002-9939(06)08398-5 Article electronically published on May 18, 2006 AN ASCENDING HNN EXTENSION OF A FREE GROUP INSIDE SL2 C DANNY CALEGARI AND NATHAN M. DUNFIELD (Communicated by Ronald A. Fintushel) Abstract. We give an example of a subgroup of SL2 C which is a strictly ascending HNN extension of a non-abelian finitely generated free group F .In particular, we exhibit a free group F in SL2 C of rank 6 which is conjugate to a proper subgroup of itself. This answers positively a question of Drutu and Sapir (2005). The main ingredient in our construction is a specific finite volume (non-compact) hyperbolic 3-manifold M which is a surface bundle over the circle. In particular, most of F comes from the fundamental group of a surface fiber. A key feature of M is that there is an element of π1(M)inSL2 C with an eigenvalue which is the square root of a rational integer. We also use the Bass-Serre tree of a field with a discrete valuation to show that the group F we construct is actually free. 1. Introduction Suppose φ: F → F is an injective homomorphism from a group F to itself. The associated HNN extension H = G, t tgt−1 = φ(g)forg ∈ G is said to be ascending; this extension is called strictly ascending if φ is not onto. In [DS], Drutu and Sapir give examples of residually finite 1-relator groups which are not linear; that is, they do not embed in GLnK for any field K and dimension n. -
Bounded Depth Ascending HNN Extensions and Π1-Semistability at ∞ Arxiv:1709.09140V1
Bounded Depth Ascending HNN Extensions and π1-Semistability at 1 Michael Mihalik June 29, 2021 Abstract A 1-ended finitely presented group has semistable fundamental group at 1 if it acts geometrically on some (equivalently any) simply connected and locally finite complex X with the property that any two proper rays in X are properly homotopic. If G has semistable fun- damental group at 1 then one can unambiguously define the funda- mental group at 1 for G. The problem, asking if all finitely presented groups have semistable fundamental group at 1 has been studied for over 40 years. If G is an ascending HNN extension of a finitely pre- sented group then indeed, G has semistable fundamental group at 1, but since the early 1980's it has been suggested that the finitely pre- sented groups that are ascending HNN extensions of finitely generated groups may include a group with non-semistable fundamental group at 1. Ascending HNN extensions naturally break into two classes, those with bounded depth and those with unbounded depth. Our main theorem shows that bounded depth finitely presented ascending HNN extensions of finitely generated groups have semistable fundamental group at 1. Semistability is equivalent to two weaker asymptotic con- arXiv:1709.09140v1 [math.GR] 26 Sep 2017 ditions on the group holding simultaneously. We show one of these conditions holds for all ascending HNN extensions, regardless of depth. We give a technique for constructing ascending HNN extensions with unbounded depth. This work focuses attention on a class of groups that may contain a group with non-semistable fundamental group at 1. -
Combinatorial Group Theory
Combinatorial Group Theory Charles F. Miller III March 5, 2002 Abstract These notes were prepared for use by the participants in the Workshop on Algebra, Geometry and Topology held at the Australian National University, 22 January to 9 February, 1996. They have subsequently been updated for use by students in the subject 620-421 Combinatorial Group Theory at the University of Melbourne. Copyright 1996-2002 by C. F. Miller. Contents 1 Free groups and presentations 3 1.1 Free groups . 3 1.2 Presentations by generators and relations . 7 1.3 Dehn’s fundamental problems . 9 1.4 Homomorphisms . 10 1.5 Presentations and fundamental groups . 12 1.6 Tietze transformations . 14 1.7 Extraction principles . 15 2 Construction of new groups 17 2.1 Direct products . 17 2.2 Free products . 19 2.3 Free products with amalgamation . 21 2.4 HNN extensions . 24 3 Properties, embeddings and examples 27 3.1 Countable groups embed in 2-generator groups . 27 3.2 Non-finite presentability of subgroups . 29 3.3 Hopfian and residually finite groups . 31 4 Subgroup Theory 35 4.1 Subgroups of Free Groups . 35 4.1.1 The general case . 35 4.1.2 Finitely generated subgroups of free groups . 35 4.2 Subgroups of presented groups . 41 4.3 Subgroups of free products . 43 4.4 Groups acting on trees . 44 5 Decision Problems 45 5.1 The word and conjugacy problems . 45 5.2 Higman’s embedding theorem . 51 1 5.3 The isomorphism problem and recognizing properties . 52 2 Chapter 1 Free groups and presentations In introductory courses on abstract algebra one is likely to encounter the dihedral group D3 consisting of the rigid motions of an equilateral triangle onto itself. -
Homology of Hom Complexes
Rose-Hulman Undergraduate Mathematics Journal Volume 13 Issue 1 Article 10 Homology of Hom Complexes Mychael Sanchez New Mexico State University, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Sanchez, Mychael (2012) "Homology of Hom Complexes," Rose-Hulman Undergraduate Mathematics Journal: Vol. 13 : Iss. 1 , Article 10. Available at: https://scholar.rose-hulman.edu/rhumj/vol13/iss1/10 Rose- Hulman Undergraduate Mathematics Journal Homology of Hom Complexes Mychael Sancheza Volume 13, No. 1, Spring 2012 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 Email: [email protected] a http://www.rose-hulman.edu/mathjournal New Mexico State University Rose-Hulman Undergraduate Mathematics Journal Volume 13, No. 1, Spring 2012 Homology of Hom Complexes Mychael Sanchez Abstract. The hom complex Hom (G; K) is the order complex of the poset com- posed of the graph multihomomorphisms from G to K. We use homology to provide conditions under which the hom complex is not contractible and derive a lower bound on the rank of its homology groups. Acknowledgements: I acknowledge my advisor, Dr. Dan Ramras and the National Science Foundation for funding this research through Dr. Ramras' grants, DMS-1057557 and DMS-0968766. Page 212 RHIT Undergrad. Math. J., Vol. 13, No. 1 1 Introduction In 1978, László Lovász proved the Kneser Conjecture [8]. His proof involves associating to a graph G its neighborhood complex N (G) and using its topological properties to locate obstructions to graph colorings. This proof illustrates the fruitful interaction between graph theory, combinatorics and topology. -
Beyond Serre's" Trees" in Two Directions: $\Lambda $--Trees And
Beyond Serre’s “Trees” in two directions: Λ–trees and products of trees Olga Kharlampovich,∗ Alina Vdovina† October 31, 2017 Abstract Serre [125] laid down the fundamentals of the theory of groups acting on simplicial trees. In particular, Bass-Serre theory makes it possible to extract information about the structure of a group from its action on a simplicial tree. Serre’s original motivation was to understand the structure of certain algebraic groups whose Bruhat–Tits buildings are trees. In this survey we will discuss the following generalizations of ideas from [125]: the theory of isometric group actions on Λ-trees and the theory of lattices in the product of trees where we describe in more detail results on arithmetic groups acting on a product of trees. 1 Introduction Serre [125] laid down the fundamentals of the theory of groups acting on sim- plicial trees. The book [125] consists of two parts. The first part describes the basics of what is now called Bass-Serre theory. This theory makes it possi- ble to extract information about the structure of a group from its action on a simplicial tree. Serre’s original motivation was to understand the structure of certain algebraic groups whose Bruhat–Tits buildings are trees. These groups are considered in the second part of the book. Bass-Serre theory states that a group acting on a tree can be decomposed arXiv:1710.10306v1 [math.GR] 27 Oct 2017 (splits) as a free product with amalgamation or an HNN extension. Such a group can be represented as a fundamental group of a graph of groups. -
Torsion, Torsion Length and Finitely Presented Groups 11
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Apollo TORSION, TORSION LENGTH AND FINITELY PRESENTED GROUPS MAURICE CHIODO AND RISHI VYAS Abstract. We show that a construction by Aanderaa and Cohen used in their proof of the Higman Embedding Theorem preserves torsion length. We give a new construction showing that every finitely presented group is the quotient of some C′(1~6) finitely presented group by the subgroup generated by its torsion elements. We use these results to show there is a finitely presented group with infinite torsion length which is C′(1~6), and thus word-hyperbolic and virtually torsion-free. 1. Introduction It is well known that the set of torsion elements in a group G, Tor G , is not necessarily a subgroup. One can, of course, consider the subgroup Tor1 G generated by the set of torsion elements in G: this subgroup is always normal( ) in G. ( ) The subgroup Tor1 G has been studied in the literature, with a particular focus on its structure in the context of 1-relator groups. For example, suppose G is presented by a 1-relator( ) presentation P with cyclically reduced relator Rk where R is not a proper power, and let r denote the image of R in G. Karrass, Magnus, and Solitar proved ([17, Theorem 3]) that r has order k and that every torsion element in G is a conjugate of some power of r; a more general statement can be found in [22, Theorem 6]. As immediate corollaries, we see that Tor1 G is precisely the normal closure of r, and that G Tor1 G is torsion-free. -
Constructing Covers of the Rose
Constructing Covers of the rose. Robert Kropholler April 25, 2017 This was discussed in class but does not appear in any of the lecture notes. I will outline the constructions and proof here. Definition 0.1. Let F (X) be the free group on the set X. Assume jXj = n. Let H be any subgroup of F (X), we define the Schrier graph G(H) as follows: • The vertices of G(H) are the left cosets of H in G. • Two cosets gH; g0H are joined by an edge if there exists an x 2 X such that xgH = g0H. It should be noted that the Schrier graph of a normal subgroup N is the Cayley graph of G=N with respect to the generating set X. Let Rn be a cell complex with 1 vertex and n edges. We identify the fundamental group of Rn with F (X). Pick a bijection between the edges of X and give each edge an orientation. Proposition 0.2. The graph G(H) is a cover of Rn. Proof. We can label each edge with an element of X and give it an orientation so that the edge (gH; xgH) points towards xgH. We can then define a map p : G(H) ! Rn. This sends each vertex of G(H) to the unique vertex of Rn and each edge to the edge with same label by an orientation preserving homeomorphism. This defines a covering map. We must check that every point has a neighbourhood such that p is a homeomorphism on this neighbourhood. -
Compressed Word Problems in HNN-Extensions and Amalgamated Products
Compressed word problems in HNN-extensions and amalgamated products Niko Haubold and Markus Lohrey Institut f¨ur Informatik, Universit¨atLeipzig {haubold,lohrey}@informatik.uni-leipzig.de Abstract. It is shown that the compressed word problem for an HNN- −1 extension hH,t | t at = ϕ(a)(a ∈ A)i with A finite is polynomial time Turing-reducible to the compressed word problem for the base group H. An analogous result for amalgamated free products is shown as well. 1 Introduction Since it was introduced by Dehn in 1910, the word problem for groups has emerged to a fundamental computational problem linking group theory, topol- ogy, mathematical logic, and computer science. The word problem for a finitely generated group G asks, whether a given word over the generators of G represents the identity of G, see Section 2 for more details. Dehn proved the decidability of the word problem for surface groups. On the other hand, 50 years after the appearance of Dehn’s work, Novikov and independently Boone proved the exis- tence of a finitely presented group with undecidable word problem, see [10] for references. However, many natural classes of groups with decidable word prob- lem are known, as for instance finitely generated linear groups, automatic groups and one-relator groups. With the rise of computational complexity theory, also the complexity of the word problem became an active research area. This devel- opment has gained further attention by potential applications of combinatorial group theory for secure cryptographic systems [11]. In order to prove upper bounds on the complexity of the word problem for a group G, a “compressed” variant of the word problem for G was introduced in [6, 7, 14]. -
Moduli Spaces of Colored Graphs
MODULI SPACES OF COLORED GRAPHS MARKO BERGHOFF AND MAX MUHLBAUER¨ Abstract. We introduce moduli spaces of colored graphs, defined as spaces of non-degenerate metrics on certain families of edge-colored graphs. Apart from fixing the rank and number of legs these families are determined by var- ious conditions on the coloring of their graphs. The motivation for this is to study Feynman integrals in quantum field theory using the combinatorial structure of these moduli spaces. Here a family of graphs is specified by the allowed Feynman diagrams in a particular quantum field theory such as (mas- sive) scalar fields or quantum electrodynamics. The resulting spaces are cell complexes with a rich and interesting combinatorial structure. We treat some examples in detail and discuss their topological properties, connectivity and homology groups. 1. Introduction The purpose of this article is to define and study moduli spaces of colored graphs in the spirit of [HV98] and [CHKV16] where such spaces for uncolored graphs were used to study the homology of automorphism groups of free groups. Our motivation stems from the connection between these constructions in geometric group theory and the study of Feynman integrals as pointed out in [BK15]. Let us begin with a brief sketch of these two fields and what is known so far about their relation. 1.1. Background. The analysis of Feynman integrals as complex functions of their external parameters (e.g. momenta and masses) is a special instance of a very gen- eral problem, understanding the analytic structure of functions defined by integrals. That is, given complex manifolds X; T and a function f : T ! C defined by Z f(t) = !t; Γ what can be deduced about f from studying the integration contour Γ ⊂ X and the integrand !t 2 Ω(X)? There is a mathematical treatment for a class of well-behaved cases [Pha11], arXiv:1809.09954v3 [math.AT] 2 Jul 2019 but unfortunately Feynman integrals are generally too complicated to answer this question thoroughly. -
Knot Theory and the Alexander Polynomial
Knot Theory and the Alexander Polynomial Reagin Taylor McNeill Submitted to the Department of Mathematics of Smith College in partial fulfillment of the requirements for the degree of Bachelor of Arts with Honors Elizabeth Denne, Faculty Advisor April 15, 2008 i Acknowledgments First and foremost I would like to thank Elizabeth Denne for her guidance through this project. Her endless help and high expectations brought this project to where it stands. I would Like to thank David Cohen for his support thoughout this project and through- out my mathematical career. His humor, skepticism and advice is surely worth the $.25 fee. I would also like to thank my professors, peers, housemates, and friends, particularly Kelsey Hattam and Katy Gerecht, for supporting me throughout the year, and especially for tolerating my temporary insanity during the final weeks of writing. Contents 1 Introduction 1 2 Defining Knots and Links 3 2.1 KnotDiagramsandKnotEquivalence . ... 3 2.2 Links, Orientation, and Connected Sum . ..... 8 3 Seifert Surfaces and Knot Genus 12 3.1 SeifertSurfaces ................................. 12 3.2 Surgery ...................................... 14 3.3 Knot Genus and Factorization . 16 3.4 Linkingnumber.................................. 17 3.5 Homology ..................................... 19 3.6 TheSeifertMatrix ................................ 21 3.7 TheAlexanderPolynomial. 27 4 Resolving Trees 31 4.1 Resolving Trees and the Conway Polynomial . ..... 31 4.2 TheAlexanderPolynomial. 34 5 Algebraic and Topological Tools 36 5.1 FreeGroupsandQuotients . 36 5.2 TheFundamentalGroup. .. .. .. .. .. .. .. .. 40 ii iii 6 Knot Groups 49 6.1 TwoPresentations ................................ 49 6.2 The Fundamental Group of the Knot Complement . 54 7 The Fox Calculus and Alexander Ideals 59 7.1 TheFreeCalculus ............................... -
NON-POSITIVELY CURVED CUBE COMPLEXES Henry Wilton Last Updated: June 29, 2012
Cube complexes Winter 2011 NON-POSITIVELY CURVED CUBE COMPLEXES Henry Wilton Last updated: June 29, 2012 1 Some basics of topological and geometric group theory 1.1 Presentations and complexes Let Γ be a discrete group, defined by a presentation P = hai j rji, say, or as the fundamental group of a connected CW-complex X. 0 1 Remark 1.1. Let XP be the CW-complex with a single 0-cell E , one 1-cell Ei 2 for each ai (oriented accordingly), and one 2-cell Ej for each rj, with attaching 2 (1) map @Ej ! XP that reads off the word rj in the generators faig. Then, by the Seifert{van Kampen Theorem, the fundamental group of XP is the group presented by P. Conversely, restricting attention to X(2) and contracting a maximal tree in (1) X , we obtain XP for some presentation P of Γ. So we see that these two situations are equivalent. We will call XP a pre- sentation complex for Γ. Here are two typical questions that we might want to be able to answer about Γ. ∗ Question 1.2. Can we compute the homology H∗(Γ) or cohomology H (Γ)? Question 1.3. Can we solve the word problem in Γ? That is, is there an algorithm to determine whether a word in the generators represents the trivial element in Γ? 1.2 Non-positively curved manifolds In general, the answer to both of these questions is `no'. Novikov (1955) and Boone (1959) constructed a finitely presentable group with an unsolvable word problem, and Cameron Gordon proved that the rank of H2(Γ) is not computable. -
1 SL2(R) Problems 1 2 1
1 SL2(R) problems 1 2 1. Show that SL2(R) is homeomorphic to S R . 2 × Hint 1: SL2(R) acts on R 0 transitively with stabilizer of (1, 0) equal − to the set of upper triangular matrices with 1’s on the diagonal, which is homeomorphic to R. This gives a “principal bundle”, i.e. a map 2 SL2(R) R 0 whose point inverses are lines. Construct a section 2 → − R 0 SL2(R) and then a homeomorphism to the product. − → Hint 2: Same thing, but for the action of SL2(R) on the upper half plane, with point inverses circles. Better yet (but uses more sophisticated math): PSL2(R)= SL2(R)/ I can be identified with the unit tangent bundle 2 ± 2 T1H of the hyperbolic plane H , which in turn can be identified with 2 1 H S∞, product with the circle at infinity. This shows that PSL2(R) × 1 2 is homeomorphic to S R , but SL2(R) is a double cover of PSL2(R). × 2. We have seen that trace Tr(A) determines the conjugacy class of A ∈ SL2(R) provided Tr(A) > 2. Show that this conjugacy class, as a subset | | 1 of SL2(R), is a closed subset homeomorphic to S R. × 3. The set of parabolics is not closed and consists of four conjugacy classes (two with trace 2 and two with trace 2), each homeomorphic to S1 R. − × 4. There are two conjugacy classes of elements of SL2(R) whose trace is a given number in ( 2, 2), each closed and homeomorphic to R2.