Geometric Group Theory

Cornelia Drut¸u

Oxford

LMS Prospects in Mathematics

Cornelia Drut¸u (Oxford) Geometric Group Theory LMS Prospects in Mathematics 1 / 22 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.

Cornelia Drut¸u (Oxford) Geometric Group Theory LMS Prospects in Mathematics 2 / 22 The Cayley graph

We study infinite finitely generated groups. Let G = hSi, S finite, 1 6∈ 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 distS .

multiplications to the left Lg (x) = gx are isometries. Terminology: In a metric space we call geodesic a path joining x, y and of length dist(x, y).

Cornelia Drut¸u (Oxford) Geometric Group Theory LMS Prospects in Mathematics 3 / 22 Cayley Graph of Dihedral Group D4

Cornelia Drut¸u (Oxford) Geometric Group Theory LMS Prospects in Mathematics 4 / 22 Commutators

Example Nilpotent groups.

Definition Let G be a group. The commutator of two elements h, k is

[h, k] = hkh−1k−1 .

For H, K subgroups of G,[ H, K]= the subgroup generated by [h, k] with h ∈ H, k ∈ K.

Cornelia Drut¸u (Oxford) Geometric Group Theory LMS Prospects in Mathematics 5 / 22 Nilpotent Groups

Construct inductively a sequence of normal subgroups : C 1G = G , C n+1G = [G, C nG] . The descending series G ≥ C 2G ≥ · · · ≥ C nG ≥ C n+1G ≥ ... 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+1G = {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.

Cornelia Drut¸u (Oxford) Geometric Group Theory LMS Prospects in Mathematics 6 / 22 Groups of isometries of real hyperbolic spaces

Other examples Finitely generated groups G with an action by isometries on a real n hyperbolic space H which is: n properly discontinuous: for every compact K in H , the set {g ∈ G ; gK ∩ K 6= ∅} is finite. n H /G is compact.

Cornelia Drut¸u (Oxford) Geometric Group Theory LMS Prospects in Mathematics 7 / 22 A group of reflections in the hyperbolic space Lattices. Mapping Class Groups

Other examples of groups

SL(n, Z) = {A ∈ Mn(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). Homeo0(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)/Homeo0(Σ) . MCG(Σ) is finitely generated ( Dehn-Lickorish).

Cornelia Drut¸u (Oxford) Geometric Group Theory LMS Prospects in Mathematics 9 / 22 Can algebra be reconstructed from geometry?

Theorem (Bass’ Theorem) A nilpotent group G has polynomial growth:

d d C1n ≤ card{v vertex ; distS(1, v) ≤ n} ≤ C2n .

Here C1 and C2 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) d Given G and d > 0, if card{v vertex ; distS(1, v) ≤ n} ≤ n for one n ≥ n0(d) then G is virtually nilpotent.

LMS Prospects in Mathematics 10 / Cornelia Drut¸u (Oxford) Geometric Group Theory 22 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 0 0 0 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 n n compact ⇒ Cayley(G, S) quasi-isometric to X (e.g. Z and R ). LMS Prospects in Mathematics 11 / Cornelia Drut¸u (Oxford) Geometric Group Theory 22 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 G1 = G/F is a subgroup in SL(n, R) ; there exists G2 of finite index in G1 and g ∈ SL(n, R) such that −1 gG2g 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 / Cornelia Drut¸u (Oxford) Geometric Group Theory 22 Hyperbolic groups

Some groups come with an action on another metric space X . Example Finitely generated groups G with properly discontinuous actions by n n isometries on the real hyperbolic space H such that H /G is compact.

Fact n In every geodesic triangle in H , 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 / Cornelia Drut¸u (Oxford) Geometric Group Theory 22 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 → ∞ the limit of the sequence   n of rescaled metric spaces X , 1 dist is a real tree. dn

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 / Cornelia Drut¸u (Oxford) Geometric Group Theory 22 Mapping Class groups

Theorem (Bestvina-Bromberg-Fujiwara) MCG(Σ) has a quasi-isometric copy inside a product of finitely many hyperbolic spaces.

Theorem (Behrstock-Drut¸u -Sapir)   For every d → ∞ the limit of the sequence MCG(Σ), 1 dist is a n dn S tree-graded space with pieces of L1–type.

LMS Prospects in Mathematics 15 / Cornelia Drut¸u (Oxford) Geometric Group Theory 22 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 / Cornelia Drut¸u (Oxford) Geometric Group Theory 22 Uniform embeddings

Definition A uniform embedding f : G → H is a map such that

ρ(distS (g, h)) ≤ kf (g) − f (h)k ≤ CdistS (g, h) , for every g, h ∈ G, (1)

where C > 0 and ρ : R+ → R+, limx→∞ 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 / Cornelia Drut¸u (Oxford) Geometric Group Theory 22 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, Sc ) of edges with exactly one endpoint in S has at least λ · cardS elements.

LMS Prospects in Mathematics 18 / Cornelia Drut¸u (Oxford) Geometric Group Theory 22 A Ramanujan graph Expanders and embeddings

Theorem (obstruction to uniform embedding)

Let Gn be an infinite sequence of (d, λ)–expanders. The space W G cannot be embedded uniformly in a Hilbert space. n∈N n

Question How to construct expanders ?

LMS Prospects in Mathematics 20 / Cornelia Drut¸u (Oxford) Geometric Group Theory 22 Expanders, lattices, embeddings

Consider G = SL(n, Z), n ≥ 3, with a finite generating set S. Consider GN = {A ∈ SL(n, Z); A = Idn modulo N}. The Cayley graphs of quotients G/GN with generating sets πN (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 / Cornelia Drut¸u (Oxford) Geometric Group Theory 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. Drut¸u, 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 22 / Cornelia Drut¸u (Oxford) Geometric Group Theory 22