Matthias Blank, Werner Thumann Geometric Group Theory II < δ Lecture Notes University of Regensburg Summer Semester 2015 Contents 1 The Geometry of Metric Spaces 5 1.1 Basics of Metric Geometry . 5 1.1.1 Basicdefinitions ..................... 5 1.1.2 Lengths, Geodesics and Angles . 6 1.1.3 LengthSpaces ...................... 12 1.2 Model Spaces of Constant Curvature . 16 1.2.1 TheEuclideanSpace. 16 1.2.2 TheSphere ........................ 18 1.2.3 TheHyperbolicSpace . 20 1.2.4 Model Spaces of Constant Curvature . 23 1.2.5 Isometry Groups of Model Spaces . 25 1.2.6 More Models for the Hyperbolic Space . 27 1.3 Spaces of curvature bounded from above . 33 1.3.1 Basic definitions and examples . 33 1.3.2 Comparison of different curvature notions . 38 1.3.3 Basicproperties . 39 1.3.4 FlatnessTheorems . 44 1.4 TheCartan-HadamardTheorem . 47 1.4.1 Motivation . 47 1.4.2 Fundamental groups and coverings . 48 1.4.3 The main theorem and outline of the proof . 54 1.4.4 Alexandrov’s Patchwork . 59 1.4.5 Proofsofthelemmata . 60 1.5 Isometries of CAT(0) spaces . 63 1.5.1 TypesofIsometries. 63 1.5.2 Examples ......................... 68 1.5.3 Flat Torus Theorem and Applications . 72 2 Exotic groups 81 2.1 Introduction............................ 81 2.2 Thompson’sgroupF .. .. .. .. .. .. .. 82 2.2.1 Basic definitions and properties . 82 2.2.2 Higher homotopy groups and cell complexes . 85 2 CONTENTS 3 2.2.3 Finiteness properties of groups . 90 2.2.4 Methods of discrete Morse Theory . 92 2.2.5 The main theorem and outline of the proof . 93 2.2.6 Connectivity of the links . 96 2.3 Thompson’sgroupV.. .. .. .. .. .. .. 99 2.3.1 Basic definitions and properties . 99 2.3.2 SimplicityofV . 101 List of Figures 1.1 Minimal great arc between two points a and b in S2 ..... 19 1.2 The two-sheeted hyperboloid H2 ................ 21 1.3 Theperspectivicprojection . 28 1.4 Perspectivic projection for ball models . 29 1.5 The isometric identification via the Cayley transformation C2. 31 1.6 Klein and Poincar´eball model and geodesics . 32 1.7 The half-plane model and some of its geodesics. 32 1.8 Product Decomposition for CAT(0) spaces . 47 1.9 Fundamental group elements . 49 1.10 Building bridges for Cartan-Hadamard . 62 1.11 Trichotomy in Isom(H2)..................... 71 1.12 Invariant sets of isometries in H2 ................ 72 2.1 Elements of Thompson’s group F ................ 83 2.2 1-dimensional complexes . 88 2.3 Asimplicialcomplex . 89 2.4 Barycentricsubdivision . 89 2.5 A descending link of a descending link . 98 2.6 Elements of Thompson’s group V ................100 2.7 A commutator in V .......................102 2.8 Another commutator in V ....................103 2.9 Proper transpositions in V ....................104 4 CHAPTER 1 The Geometry of Metric Spaces Last semester, we have studied, in particular, two examples of metric spaces coming from geometric group theory, that is finitely generated groups with a word metric and Cayley graphs associated to such groups. In this chapter, we will explore the geometry of metric spaces in more detail. 1.1 Basics of Metric Geometry 1.1.1 Basic definitions We begin by recalling some basic concepts and fixing some notations: Definition 1.1.1. Let X be a set. A map d: X X R 0 is × −→ ≥ ∪ {∞} called a pseudo-metric, if the following holds (i) For all x X, we have d(x,x) = 0. ∈ (ii) The map d is symmetric, i.e. x,y X d(x,y)= d(y,x). ∀ ∈ (iii) The triangle inequality holds, i.e.: x,y,z X d(x, z) d(x,y)+ d(y, z). ∀ ∈ ≤ A pseudo-metric d is called a generalised metric if the following holds: (i’) For all x,y X, we have d(x,y) = 0 if and only if x = y. ∈ A generalised metric d with image in R 0 is called a metric. The pair (X, d) ≥ is then called a metric space. Notation 1.1.2. Let (X, d) be a pseudo-metric space and x X. ∈ 5 CHAPTER 1. THE GEOMETRY OF METRIC SPACES 6 (i) For any r R , we write B(x, r) := y X d(x,y) < r for the ∈ >0 { ∈ | } open r-ball around x in X. (ii) Similarly, for any r R 0, we write B¯(x, r) := y X d(x,y) r ∈ ≥ { ∈ | ≤ } for the closed r-ball around x in X. Note that in general, B¯(x, r) can be strictly larger than the closure B(x, r). Example 1.1.3 (One nice and one boring metric space). (i) For n N, we write En to denote the standard Euclidean metric R∈n n space ( , d2 ) where n dn(x,y)= (x y )2 2 v i − i u i=1 uX t for x = (x ,...,x ) Rn and y = (y ,...,y ) Rn. 1 n ∈ 1 n ∈ (ii) For any set X, there is a metric on X given by setting for all x,y X ∈ 1 if x = y d(x,y) := 6 (0 if x = y, called the discrete metric on X. General metric spaces can be rather savage and we will in general restrict our attention to well-behaved classes. For example, we will often consider metric spaces in which, like in Euclidean space, every closed bounded set is compact: Definition 1.1.4 (Locally compact and proper). Let (X, d) be a metric space. (i) We call (X, d) locally compact, if for every x X, there exists an ∈ r R , such that B¯(x, r) is compact. ∈ >0 (ii) We call (X, d) proper, if for every x X and every r R the ∈ ∈ >0 closed r-ball B¯(x, r) is compact. Remark 1.1.5. Clearly, every proper metric space is locally compact and complete. The converse does not hold, however. For example, every set with the discrete metric is locally compact and complete, but an infinite set with the discrete metric is clearly not proper. 1.1.2 Lengths, Geodesics and Angles In this section, we will discuss three elementary geometric notions, lengths of paths, geodesics and angles, in the metric setting. CHAPTER 1. THE GEOMETRY OF METRIC SPACES 7 We begin by recalling the definition of the length of a path: Definition 1.1.6 (Length of paths). (i) Let X be a topological space. A continuous map I X, where I R −→ ⊂ is a non-empty compact interval in R, is called a path in X. (ii) For a, b R with a b, we call a tuple (t ,...,t ) in R a partition ∈ ≤ 0 n+1 of [a, b] if t = a and t = b and t t for all i 0,...,n . 0 n+1 i ≤ i+1 ∈ { } (iii) Let (X, d) be a metric space and c: [a, b] X a path in X. We call −→ n l(c) := sup d(c(ti+1), c(ti)) (t0,...,tn+1) a partition of [a, b] , ni=0 o X the length of c. (iv) We call a path c: [a, b] X rectifiable if l(c) is finite. −→ Caveat 1.1.7. Not every path is rectifiable and there are path-connected metric spaces that do not admit rectifiable paths between every pair of points. Proposition 1.1.8 (Elementary properties of length). Let (X, d) be a met- ric space. (i) Let c: [a, b] X be a path in X. We have l(c) d(c(a), c(b)). −→ ≥ Furthermore, we have l(c) = 0 if and only if c is constant. (ii) Let c : [a , b ] X and c : [a , b ] X be two paths in X such 1 1 1 −→ 2 2 2 −→ that c (b )= c (a ). Let c c denote the concatenation of c and c , 1 1 2 2 1 ∗ 2 1 2 i.e. the path c c : [0, b + b a a ] X given by 1 ∗ 2 2 1 − 2 − 1 −→ c (t + a ) if t [0, b a ] t 1 1 ∈ 1 − 1 7−→ (c2(t + a1 + a2 b1) if t [b1 a1, b2 + b1 a2 a1]. − ∈ − − − Then l(c c )= l(c )+ l(c ). 1 ∗ 2 1 2 (iii) Let c: [a, b] X be a rectifiable path. Let c¯: [ b, a] X be the −→ − − −→ inverse path of c, i.e. the path given by t c( t). 7−→ − Then l(¯c)= l(c). (iv) Let c: [a, b] X be a rectifiable path. Then the arc length function −→ of c λ : [a, b] [0, l(c)] c −→ t l c 7−→ |[a,t] is continuous and weakly monotonic. CHAPTER 1. THE GEOMETRY OF METRIC SPACES 8 (v) Let c: [a, b] X be a rectifiable path. Then there exists a unique −→ rectifiable path c˜: [0, l(c)] X, s.t. c˜ λ = c and l(˜c )= t. −→ ◦ c |[0,t] (vi) Let (cn : [a, b] X)n N be a sequence of paths converging uniformly −→ ∈ to a path c: [a, b] X. If c is rectifiable, then l(c) lim infn l(cn). −→ ≤ →∞ Proof. Parts (i),(ii),(iii) and (v) are easy and left as an exercise. (iv) It is clear that λ is weakly monotonic using property (ii). Fix ε c ∈ R>0. Since [a, b] is compact, c is uniformly continuous. Hence, there exists δ R such that for all t,t [a, b] with d(t,t ) < δ, we have ∈ >0 ′ ∈ ′ d(c(t), c(t′)) < ε/2. Let (t0,...,tn+1) be a partition of [a, b], such that (a) i 0,...,n d(ti,ti+1) < δ.