Geometric Group Theory

Transcription

1 Cornelia Druţu Oxford LMS Prospects in Mathematics LMS Prospects in Mathematics 1 /

2 Groups and Structures Felix Klein (Erlangen Program): a geometry can be understood via the group of transformations preserving it. Instead of geometry: any other mathematical structure. This idea can be used in the reversed order: understand a group via its actions on some (metric) space with a good structure. LMS Prospects in Mathematics 2 /

3 The Cayley graph We study infinite finitely generated groups. Let G = S, S finite, 1 S, x S x 1 S. The Cayley graph Cayley(G, S) of G with respect to S is a non-oriented graph with: set of vertices G; edges = pairs of elements {g, h}, such that h = gs, for some s S. Cayley(G, S) is connected (because S generates G); Cayley(G, S) is a metric space: assume edges have length 1, take shortest path metric dist S. multiplications to the left L g (x) = gx are isometries. Terminology: In a metric space we call geodesic a path joining x, y and of length dist(x, y). LMS Prospects in Mathematics 3 /

4 Cayley Graph of Dihedral Group D 4 LMS Prospects in Mathematics 4 /

5 Commutators Example Nilpotent groups. Definition Let G be a group. The commutator of two elements h, k is [h, k] = hkh 1 k 1. For H, K subgroups of G, [H, K]= the subgroup generated by [h, k] with h H, k K. LMS Prospects in Mathematics 5 /

6 Nilpotent Groups Construct inductively a sequence of normal subgroups : The descending series C 1 G = G, C n+1 G = [G, C n G]. G C 2 G C n G C n+1 G... is the lower central series of the group G. Definition A group G is (k-step) nilpotent if there exists k such that C k+1 G = {1}. The minimal such k is the class of G. Examples 1 An abelian group is nilpotent of class 1. 2 The group of upper triangular n n matrices with 1 on the diagonal is nilpotent of class n 1. LMS Prospects in Mathematics 6 /

7 Groups of isometries of real hyperbolic spaces Other examples Finitely generated groups G with an action by isometries on a real hyperbolic space H n which is: properly discontinuous: for every compact K in H n, the set {g G ; gk K } is finite. H n /G is compact. LMS Prospects in Mathematics 7 /

8 A group of reflections in the hyperbolic space

9 Lattices. Mapping Class Groups Other examples of groups SL(n, Z) = {A M n (Z) ; det A = 1}. Consider Σ an orientable compact surface (with or without boundary). Homeo(Σ) = the group of homeomorphisms of Σ (i.e. invertible transformations f : Σ Σ, such that f, f 1 continuous). Homeo 0 (Σ) = the subgroup of homeomorphisms that can be connected to the identity by a continuous path of homeomorphisms. The mapping class group MCG(Σ) = the quotient Homeo(Σ)/Homeo 0 (Σ). MCG(Σ) is finitely generated ( Dehn-Lickorish). LMS Prospects in Mathematics 9 /

10 Can algebra be reconstructed from geometry? Theorem (Bass Theorem) A nilpotent group G has polynomial growth: C 1 n d card{v vertex ; dist S (1, v) n} C 2 n d. Here C 1 and C 2 depend on the generating set S, d depends only on G. Theorem (M. Gromov) If G has polynomial growth then G is virtually nilpotent (i.e. has a nilpotent subgroup of finite index). Theorem (Y. Shalom, T. Tao) Given G and d > 0, if card{v vertex ; dist S (1, v) n} n d for one n n 0 (d) then G is virtually nilpotent. LMS Prospects in Mathematics 10 /

11 Quasi-isometries Some groups can be entirely recognized from their Cayley graphs. Definition (a loose geometric equivalence) An (L, C) quasi-isometry is a map f : X Y such that: 1 L dist(x, x ) C dist(f (x), f (x )) Ldist(x, x ) + C every point in Y is at distance at most C from a point in f (X ). X and Y are quasi-isometric. Example A group and a finite index subgroup; or a quotient by a finite normal subgroup. Example G acts properly discontinuously on a metric space X such that X /G is compact Cayley(G, S) quasi-isometric to X (e.g. Z n and R n ). LMS Prospects in Mathematics 11 /

12 Theorems of Rigidity Theorem If a group G is quasi-isometric to SL(n, Z), n 3, then: there exists F finite normal subgroup in G such that G 1 = G/F is a subgroup in SL(n, R) ; there exists G 2 of finite index in G 1 and g SL(n, R) such that gg 2 g 1 has finite index in SL(n, Z). (A. Eskin) A similar result for MCG(Σ). (J. Behrstock-B. Kleiner-Y. Minsky- L. Mosher). LMS Prospects in Mathematics 12 /

13 Hyperbolic groups Some groups come with an action on another metric space X. Example Finitely generated groups G with properly discontinuous actions by isometries on the real hyperbolic space H n such that H n /G is compact. Fact In every geodesic triangle in H n, each edge is contained in the tubular neighbourhood of radius ln 3 of the union of the other two edges. For every group in the Example, the same is true in every Cayley graph of G with ln 3 replaced by a constant δ depending on S. Such a group is called a hyperbolic group. Similar terminology for metric spaces. LMS Prospects in Mathematics 13 /

14 Hyperbolic spaces are everywhere Note: A hyperbolic metric space is a perturbation of a tree. Theorem X is hyperbolic if and only( if for every ) d n the limit of the sequence of rescaled metric spaces X, 1 d n dist is a real tree. Several good reasons to be interested in hyperbolic groups and spaces: Random groups are hyperbolic (M. Gromov). Given a surface Σ as above, with genus at least 2, its curve complex C(Σ) is hyperbolic. (H. Masur- Y. Minsky). This complex has: vertices corresponding to homotopy classes of simple closed curves; two vertices are joined by an edge if the two classes have disjoint representatives. LMS Prospects in Mathematics 14 /

15 Mapping Class groups Theorem (Bestvina-Bromberg-Fujiwara) MCG(Σ) has a quasi-isometric copy inside a product of finitely many hyperbolic spaces. Theorem (Behrstock-Druţu -Sapir) ) For every d n the limit of the sequence (MCG(Σ), 1 dist dn S is a tree-graded space with pieces of L 1 type. LMS Prospects in Mathematics 15 /

16 Embeddings in Hilbert spaces Trick from theoretical computer science and combinatorial optimisation: To solve a problem embed the combinatorial structure in a well understood metric space (an Euclidean space); use the ambient geometry to devise an algorithm. For infinite groups, the embeddings must be in Hilbert spaces. Open Question (Cornulier-Tessera-Valette) The only f.g. groups with quasi-isometric copies in Hilbert spaces are Abelian groups. Proved for nilpotent groups (Pansu-Semmes). LMS Prospects in Mathematics 16 /

17 Uniform embeddings Definition A uniform embedding f : G H is a map such that ρ(dist S (g, h)) f (g) f (h) Cdist S (g, h), for every g, h G, (1) where C > 0 and ρ : R + R +, lim x ρ(x) =. Theorem (Guoliang Yu) A group with a uniform embedding in a Hilbert space satisfies the Novikov conjecture and the coarse Baum-Connes conjecture. LMS Prospects in Mathematics 17 /

18 Expanders Question Maybe all f.g. groups admit a uniform embedding in a Hilbert space? Definition A (d, λ) expander is a finite graph Γ: of valence d in every vertex; such that for every set S containing at most half of the vertices, the set E(S, S c ) of edges with exactly one endpoint in S has at least λ cards elements. LMS Prospects in Mathematics 18 /

19 A Ramanujan graph

20 Expanders and embeddings Theorem (obstruction to uniform embedding) Let G n be an infinite sequence of (d, λ) expanders. The space n N G n cannot be embedded uniformly in a Hilbert space. Question How to construct expanders? LMS Prospects in Mathematics 20 /

21 Expanders, lattices, embeddings Consider G = SL(n, Z), n 3, with a finite generating set S. Consider G N = {A SL(n, Z) ; A = Id n modulo N}. The Cayley graphs of quotients G/G N with generating sets π N (S) compose an infinite sequence of (d, λ) expanders. Relevant property of SL(n, Z), n 3: the property (T) of Kazhdan. Theorem (Gromov, Arzhantseva-Delzant) The exist f.g. groups with a family of expanders quasi-isometrically embedded in a Cayley graph. Proof uses random groups. The group is a direct limit of hyperbolic quotients. LMS Prospects in Mathematics 21 /

22 GGT People in UK Cambridge: J. Button, D. Calegari Durham: J. Parker, N. Peyerimhoff Edinburgh, Heriot Watt University: J. Howie Glasgow: T. Brendle, P. Kropholler, S. Pride Liverpool: Mary Rees London (U. College London): H. Wilton. Newcastle: Sarah Rees, A. Vdovina Oxford: M. Bridson, C. Druţu, M. Lackenby, P. Papasoglu. Southampton: I. Leary, A. Martino, A. Minasyan, G. Niblo, B. Nucinkis Warwick: B. Bowditch, S. Schleimer, C. Series LMS Prospects in Mathematics /