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Automorphism Groups of Combinatorial Structures Automorphism groups of combinatorial structures Anne Thomas Notes prepared by Ben Brawn, Tim Bywaters and Thomas Taylor Abstract This is a series of lecture notes taken by students during a 5 lecture se- ries presented by Anne Thomas in 2016 at the MATRIX workshop: The Winter of Disconnectedness. Acknowledgements We would like to thank John J. Harrison for kindly sharing his notes. We would also like to thank MATRIX for funding and hosting the Winter of Disconnectedness work- shop where these lectures were presented. 1 Introduction These lectures are based on the survey paper [20] of Farb, Hruska and Thomas. We give additional background in these lectures and update some results, including some answers to questions posed in [20]. Both the survey [20] and these lectures were inspired by the paper [31], which explores the theory of tree lattices and relates results to the classical case. The primary goal of these lectures was to present the main examples of polyhedral complexes and then use them to generate interesting examples of locally compact groups. Results are stated about these groups with a focus on lattices and comparisons with the well developed theory of Lie groups. Some results will be identical to results from Lie groups with identical proofs, some will require different techniques to prove and some will directly contrast. Using this point of view, it is possible to pull insight from one case to another, thus it is possible to gain insight into Lie groups by studying groups acting on polyhedral complexes. When possible, at the start of each section we give references which provide the Anne Thomas The University of Sydney School of Mathematics and Statistics, Carslaw Building NSW 2006 e-mail: anne.thomas@sydney.edu.au 1 2 Anne Thomas reader with a deeper background into the topic which is beneficial but not required. The interested reader may want to read further than what is presented here. In that case we recommend the survey [20] which presents a long, but motivated, list of research problems. For the remainder of this section we define a notion of curvature on metric spaces and consider the basics of isometries on these space. We then give a short introduc- tion to lattices and provide elementary examples. We finish with a rough intuition for symmetric spaces. This intuition is not meant to be a definition, but instead places future results in context. In Section 2 we standardise the notation we will be using for graphs and specif- ically trees. We then provide some results on tree lattices. In Section 3 we define a general polyhedral complex and see how previous examples fit this mould. We then explore results related to buildings which are an important example. Finally in Section 4 we present examples of polyhedral complexes which have received recent attention in the literature. 1.1 Models of metric spaces In this section we introduce the concept of metric spaces. We then define 3 funda- mental examples of metric spaces. These are the unique n-dimensional Riemannian manifolds with constant sectional curvature 1, 0 and −1. In later sections, we will compare arbitrary metric spaces with these to define a notion of curvature in a gen- eral geodesic metric space. We suggest [10] as a very complete reference for this section. Definition 1. Suppose (X;d) is a metric space. A geodesic segment is a function g : [a;b] ⊂ R ! X such that for all s;t 2 [a;b] we have d(g(s);g(t)) = js −tj: (1) A geodesic ray is a function g : [a;¥) ! X such that for all s;t 2 [a;¥) equation (1) holds. A geodesic line is a function g : (−¥;¥) ! X such that for all s;t 2 (−¥;¥) equation (1) holds. Remark 1. We will often refer to a geodesic segment, ray or line by just geodesic when the context is clear. We also identify a geodesic g with its image in X. Example 1. Here we provide the examples of the 3 unique n-dimensional Rieman- nian manifolds with constant curvature 1, 0 and −1 respectively. Instead of defining the metric explicitly, it is equivalent to describe the geodesics in the space, as they determine the metric. n n • Let S denote the unit sphere in R with the induced Euclidean metric. Then all geodesics are arcs of great circles. If two points are not antipodal then there is a unique geodesic connecting them. Alternatively there are infinitely many geodesics between two antipodal points. Automorphism groups of combinatorial structures 3 n • Let E denote n-dimensional Euclidean space. We make the distinction between n n n E and R ; E is not equipped with a vector space structure, nor is there a fixed origin. This allows us to work in a coordinate free manner. The geodesics in Euclidean space are straight lines. n • Let H denote n-dimensional real hyperbolic space. There are two models of hyperbolic space that we will use when convenient. – Consider the half plane model U = fz 2 C : Á(z) > 0g for H2. Then geodesics take the form of vertical lines or segments of semicir- cles perpendicular to the Real axis. These are shown in Figure 1. Fig. 1 Geodesics in the upper half plane model of hyperbolic space Im · i Re – An alternative model H2 is the Poincare´ disk D. This is given by D = fx 2 C : jxj < 1g: Geodesics in the disk are either diameters or arcs of circles perpendicular to the boundary of the unit disk. These are shown in Figure 2. There is an isom- etry U ! D which wraps the Real axis into a circle with endpoints meeting at the point at infinity. 1.2 Curvature condition and isometries Given the spaces with constant sectional curvature described above we describe a notion of curvature in a general geodesic space. We focus on spaces with non- 4 Anne Thomas Fig. 2 Geodesics in the Poincare´ disk model of hyperbolic space positive curvature. As we will see later, these spaces appear naturally in many set- tings. Again, [10] is a good reference for this section. Definition 2. For a metric space (X;d) we define the following: • We say (X;d) is geodesic if any two points can be connected by a geodesic. Note that we do not require this geodesic to be unique. • A geodesic triangle D(x1;x2;x3) between points x1;x2;x3 2 X is a union [ f[xi;x j] : i; j 2 f1;2;3gg where [xi;x j] is a geodesic from xi to x j. A comparison triangle D(x¯1;x¯2;x¯3) for 2 D(x1;x2;x3) in E is a union of geodesics [ f[x¯i;x¯j] : i; j 2 f1;2;3gg 2 where [x¯i;x¯j] is the unique geodesic between pointsx ¯i;x¯j 2 E which are chosen to satisfy d(xi;x j) = d(x¯i;x¯j). If p 2 [xi;x j] ⊂ D(x1;x2;x3), then a comparison point for p is the uniquep ¯ 2 [x¯i;x¯j] with d(p¯;x¯i) = d(p;xi). • We say D(x1;x2;x3) satisfies the CAT(0) inequality if for any p;q 2 D(x1;x2;x3) we have d(p;q) ≤ d(p¯;q¯): Note that satisfying the CAT(0) inequality is independent of choice of compari- son triangle. • We call X a CAT(0) space if every geodesic triangle in X satisfies the CAT(0) inequality. Automorphism groups of combinatorial structures 5 Fig. 3 On the left we have a triangle in a CAT(0) space and on the right a comparison triangle in Euclidean space. x¯2 x2 x¯1 x 1 x¯3 x3 Remark 2. Similarly we can define CAT(−1) and CAT(1) by comparing with H2 and S2 respectively. Because the sphere has finite diameter and so geodesics have finite length, to show a space is CAT(1) it only makes sense to compare triangles with diameters less than 2p [10, Part II]. It can also be seen that CAT(−1) implies CAT(0) and CAT(0) implies CAT(1) [10, Theorem 1.12]. n Example 2. It is easy to see that E is CAT(0) for all n 2 N. More generally a normed vector space is CAT(0) if and only if it is an inner product space, for a proof see [10, Proposition 1.14]. This shows that any Banach space that is not a Hilbert space is not CAT(0). Proposition 1. Suppose X is a CAT(0) metric space. Then there exists a unique geodesic between any two distinct points. Proof. The following argument is presented pictorially in Figure 4. Suppose x1 and x2 are two points and g1 and g2 are two geodesics from x1 to x2. Then for any p1 2 g1 there exists a unique p2 2 g2 such that d(x1; p1) = d(x1; p2). We will show p1 = p2. Consider the triangle D(x1; p1;x2) which is given by g1 [g2. Taking a comparison triangle D(x¯1; p¯1;x¯2) we must have d(x¯1; p¯1) + d(x¯2; p¯1) = d(x¯1;x¯2): This can only happen ifp ¯1 is on the unique geodesic fromx ¯1 tox ¯2. This shows D(x¯1; p¯1;x¯2) is in fact a line. Taking a comparison pointp ¯2 for p2, we must have p¯2 = p¯1. Applying the CAT(0) inequality we have d(p1; p2) ≤ d(p¯1; p¯2) = 0: This shows g1 ⊂ g2.
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