geometric group theorymaxcu/corneliadrutulmsprospects… · a group g is (k-step)nilpotentif there...

Geometric Group Theory

Cornelia Drutu

Oxford

LMS Prospects in Mathematics

Cornelia Drutu (Oxford) Geometric Group Theory LMS Prospects in Mathematics 1 / 22

Groups and Structures

Felix Klein (Erlangen Program): a geometry can be understood viathe group of transformations preserving it.

Instead of geometry: any other mathematical structure.

This idea can be used in the reversed order: understand a group viaits actions on some (metric) space with a good structure.


The Cayley graph

We study infinite finitely generated groups.Let G = 〈S〉, 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-orientedgraph 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, takeshortest 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 oflength dist(x , y).


Cayley Graph of Dihedral Group D4


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] withh ∈ H, k ∈ K .


Nilpotent Groups

Construct inductively a sequence of normal subgroups :

C 1G = G , Cn+1G = [G ,CnG ] .

The descending series

G ≥ C 2G ≥ · · · ≥ CnG ≥ Cn+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 diagonalis nilpotent of class n − 1.


Groups of isometries of real hyperbolic spaces

Other examples

Finitely generated groups G with an action by isometries on a realhyperbolic space Hn which is:

properly discontinuous: for every compact K in Hn, the set{g ∈ G ; gK ∩ K 6= ∅} is finite.

Hn/G is compact.


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 beconnected to the identity by a continuous path of homeomorphisms.The mapping class group MCG(Σ) = the quotientHomeo(Σ)/Homeo0(Σ) .MCG(Σ) is finitely generated ( Dehn-Lickorish).


Can algebra be reconstructed from geometry?

Theorem (Bass’ Theorem)

A nilpotent group G has polynomial growth:

C1nd ≤ card{v vertex ; distS(1, v) ≤ n} ≤ C2nd .

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 anilpotent subgroup of finite index).

Theorem (Y. Shalom, T. Tao)

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

Cornelia Drutu (Oxford) Geometric Group TheoryLMS Prospects in Mathematics 10 /

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:

1Ldist(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 normalsubgroup.

Example

G acts properly discontinuously on a metric space X such that X/G iscompact ⇒ Cayley(G , S) quasi-isometric to X (e.g. Zn and Rn).


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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 asubgroup in SL(n,R) ;there exists G2 of finite index in G1 and g ∈ SL(n,R) such thatgG2g−1 has finite index in SL(n,Z). (A. Eskin)

A similar result for MCG(Σ). (J. Behrstock-B. Kleiner-Y. Minsky- L.Mosher).


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Hyperbolic groups

Some groups come with an action on another metric space X .

Example

Finitely generated groups G with properly discontinuous actions byisometries on the real hyperbolic space Hn such that Hn/G is compact.

Fact

In every geodesic triangle in Hn, each edge is contained in the tubularneighbourhood 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 ofG with ln 3 replaced by a constant δ depending on S .Such a group is called a hyperbolic group. Similar terminology for metricspaces.


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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 dn →∞ the limit of the sequence

of rescaled metric spaces(

X , 1dndist

)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 complexC(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 disjointrepresentatives.


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Mapping Class groups

Theorem (Bestvina-Bromberg-Fujiwara)

MCG(Σ) has a quasi-isometric copy inside a product of finitely manyhyperbolic spaces.

Theorem (Behrstock-Drutu -Sapir)

For every dn →∞ the limit of the sequence(MCG(Σ), 1

dndistS

)is a

tree-graded space with pieces of L1–type.


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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 areAbelian groups.

Proved for nilpotent groups (Pansu-Semmes).


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Uniform embeddings

Definition

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

ρ(distS(g , h)) ≤ ‖f (g)− f (h)‖ ≤ 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 Novikovconjecture and the coarse Baum-Connes conjecture.


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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, theset E (S ,Sc) of edges with exactly one endpoint in S has at leastλ · cardS elements.


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A Ramanujan graph

Expanders and embeddings

Theorem (obstruction to uniform embedding)

Let Gn be an infinite sequence of (d , λ)–expanders.The space

∨n∈N Gn cannot be embedded uniformly in a Hilbert space.

Question

How to construct expanders ?


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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-isometricallyembedded in a Cayley graph.

Proof uses random groups.

The group is a direct limit of hyperbolic quotients.


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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. Drutu, M. Lackenby, P. Papasoglu.

Southampton: I. Leary, A. Martino, A. Minasyan, G. Niblo, B.Nucinkis

Warwick: B. Bowditch, S. Schleimer, C. Series


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