Math 697: Introduction to Geometric Group Theory Notes from course given by Dick Canary, Fall 2014. August 23, 2016 0 Overview 0.1 A Rambling Geometric group theory is really \a collection of things put together by a state of mind." These things defi- nitely include hyperbolic group theory, but also various other things such as CAT(0) groups, topological group theory, and so on ... A little more concretely, geometric group theory is the study of groups by looking at their actions on spaces. Given a [finitely-presented] group G = hx1; : : : ; xn j r1; : : : ; rmi, we can define the word metric distance by setting d(g; h) to be the minimal length of word in fx1; : : : ; xng (and their inverses) over all representations −1 of g h. We define the Cayley graph ΓG of G by taking the vertices to be the elements of G and drawing edges labelled xi joining each g to gxi ∼ Example 0.1. • If G = Z = ha ji, then ΓG is a line (= R.) a a a a ··· ··· a−2 a−1 1 a a2 2 • If G = Z = ha; b j [a:b]i, then ΓG is a 2-dimensional lattice (\what is this? It is graph paper.") ab = ba b 0 a 1 1 • If G = F2 = ha; b ji, then ΓG is an infinite 4-valent tree (the universal cover of S _ S )|see below If M is a compact manifold, then π1(M) acts properly discontinuously and co-compactly on M~ . 2 2 Example 0.2. If M = T , then π1(M) = Z , which acts properly discontinuously and cocompactly on M~ = R2; we note that R2 \looks like" Z2 (coarsely speaking.) ~ If M is a compact 3-manifold (e.g. a solid genus-2 torus), M is a thickening of ΓF2 , the infinite 4-valent tree. We make precise the notion of \looks like" by using the notion of quasi-isometry. Note that our spaces X and Y are usually assumed to be proper (i.e. closed balls are compact) and geodesic (i.e. distance between two points is given by the length of a shortest path.) 1 a−1 ba b a−1 ba−1 a ba b−1 b−1 a−1 ba a−1 ba b b b a−1 ba−1 a−1 a a ba b−1 b−1 b−1 b−1 a−1 ba a−1 a a−1 a a−1 ba b−1 b−1 b b a−1 ba−1 a ba a−1 ba−1 a ba b b−1 b−1 b−1 b−1 a−1 ba a−1 ba b b b b a−1 ba−1 a−1 a−1 a a a ba b−1 b−1 b−1 b−1 a−1 a a−1 a b−1 b−1 b−1 b−1 a−1 ba−1 a ba a−1 ba−1 a ba b−1 b−1 b−1 b−1 b−1 b−1 b−1 a−1 a a−1 ba a−1 ba a−1 a b−1 b−1 b b a−1 ba−1 a−1 a a ba b−1 b−1 b−1 b−1 a−1 a a−1 a b−1 b−1 b−1 a−1 ba−1 a ba b−1 b−1 b−1 a−1 a b−1 Figure 1: Part of a Cayley graph of a nonabelian free group 1 Definition 0.3. h : X ! Y is a quasi-isometric embedding if there exist k; c s.t. k d(x; z) − c ≤ d (h(x); h(z)) ≤ kd(x; z) + c for all x; z 2 X. h is a quasi-isometry if in addition it is \coarsely surjective", i.e. there exists D ≥ 0 s.t. for all y 2 Y , there exists x 2 X s.t. d(h(x); y) ≤ D.1 A key result in geometric group theory is the Milnor-Svarcˇ lemma, which states that if G acts properly discontinuously and co-compactly on a space X, then X is quasi-isometric to G with the word metric. Note that this notion of quasi-isometry, though seemingly loose and geometric, can capture algebraic aspects of structure (or aspects of algebraic structure.) e.g. G quasi-isometric to a free group implies G virtually free.2 There is also Gromov's theorem, which states that any group with polynomial growth is virtually nilpotent. In fact, we can also use algebra to clarify geometry, as in the following [re-]formulation of hyperbolic 1Think of the plane, and graph paper. 2\Virtually" or \almost" in this context means true up to (true for a) finite-index subgroup. These adjectives will appear a lot, because quasi-isometry fails to distinguish between the whole group and finite-index subgroups. 2 spaces by Gromov: Fact. Suppose X is simply-connected with (sectional) curvature ≤ −k < 0. Then there exists δ = δ(k) such that any geodesic triangle in X is δ-slim, meaning that if T has vertices x; y; z, then [x; z] ⊂ Nδ([x; y][[y; z]), i.e. 8t 2 [x; z]: 9s 2 [x; y] [ [y; z] s.t. d(s; t) < δ. Definition 0.4 (Gromov, possibly Alexander). X is δ-hyperbolic if every geodesic triangle in X is δ-slim. e.g. trees are 0-hyperbolic. A space is hyperbolic if it is δ-hyperbolic for some δ ≥ 0. A group is hyperbolic if it acts properly discontinuously and co-compactly on a hyperbolic space. Hyperbolic groups have some very nice properties: • Hyperbolic groups have solvable word problem (which we might formulate as \can I build the Cayley graph [algorithmically]?") by Dehn's algorithm. • They also have solvable conjugacy problem and isomorphism problem.3 • The Tits Alternative: if a hyperbolic group G is not virtually cyclic, it contains a F2. • A hyperbolic group cannot contain a copy of Z2. • A hyperbolic group has only finitely many conjugacy classzes of finite subgroups. • Being a hyperbolic space / group is a condition invariant under quasi-isometry. Hyperbolic spaces are those with constant negative curvature (-1); we can generalize this notion to look at spaces X with constant non-positive curvature, and this brings us to the theory of CAT(0) groups: Definition 0.5. A simply-connected space X is CAT(0) if every geodesic triangle in X is at least as thin as a Euclidean triangle with the same edge lengths. G is CAT(0) if it acts properly and cocompactly on a CAT(0) space. Definition 0.6. CAT(0) cube complexes are formed by gluing Euclidean unit cubes along faces. e.g. R2 is a CAT(0) cube complex Example 0.7. Right-Angled Artin Groups (RAAGs) are CAT(0) groups AΓ formed as follows: start with a finite graph Γ; define (0) (1) AΓ = hΓ j vw = wv () (v; w) ⊂ Γ i i.e. the elements of AΓ are the vertices of Γ, and we add a commutator relation between two elements iff there is an edge between the corresponding vertices. e.g. if Γ is the empty graph on n vertices, then AΓ = Fn. n If Γ = Kn, then AΓ = Z . 2 2 If Γ consists of two disjoint edges on 4 vertices, then AΓ = Z ∗ Z . These groups are \interpolating between the free group and the free abelian group." Theorem 0.8 (Agol, Wise). If a hyperbolic group G acts properly discontinuously and cocompactly on a CAT(0) cube complex (in this case we say G is a cubulation), then a finite-index subgroup of G embeds in a RAAG. Theorem 0.9 (Haglund-Wise, Kahn-Markovic). If M is a closed hyperbolic 3-manifold, then M is cubulated. Finally, there is topological group theory. The prototypical result here is the proposition that states that every subgroup of a free group is free, proven using covering spaces. There is also the following generalisation of this: Theorem 0.10 (Kurosh). If G = G1 ∗ G2 and H ⊂ G, then H = F ∗λ2Λ Hλ, where F is free, and each HΛ is conjugate to a a subgroup of G1 or G2. (See Figure 2.) 3Something involving Tietze transformations was name-dropped here. 3 Figure 2: A free product may be thought of as the fundamental group of a wedge of spaces. This has the following corollary: Corollary. If G is finitely-generated, then G has a unique free decomposition G = G1 ∗ · · · ∗ Gn where each Gi is freely indecomposable (i.e. does not split as a free product.) Other topics that might come under this last heading include JSJ decompositions and Bass-Serre theory. 0.2 Definitions and quasi-isometry Definition 0.11. Given X a metric space and α : [0; 1] ! X a path, n−1 X `X (α) := sup d(α(ti); α(ti+1)) 0=t <t <···<t =1 0 1 n i=0 Definition 0.12. α is rectifiable if `X (α) < 1. Definition 0.13. α is geodesic if `X (α) = d(α(0); α(1)). Exercise. (1) A subpath of a geodesic is a geodesic, (2) If 0 ≤ s ≤ t ≤ u ≤ 1, then d(α(s); α(t)) + d(α(t); α(u)) = d(α(s); α(u)) if α is a geodesic. Definition 0.14. A metric space is geodesic if any two points are joined by a (by some) geodesic. e.g. Cayley graphs, Riemannian manifolds, curve complexes, etc. Definition 0.15. X is proper if all closed metric balls are compact. Our spaces will be proper geodesic metric spaces (mostly.) We always assume that G acts by isometries on X, i.e.