cayley graphs of finitely generated groupspborod/ystw/simon2.pdf · cayley graphs of finitely...

Cayley Graphs of Finitely Generated Groups

Simon Thomas

Rutgers University

13th May 2014

Simon Thomas (Rutgers University) 7th Young Set Theory Workshop 13th May 2014

Cayley graphs of finitely generated groups

DefinitionLet G be a f.g. group and let S ⊆ G r {1 } be a finite generating set.Then the Cayley graph Cay(G,S ) is the graph with vertex set G andedge set

E = { {x , y} | y = xs for some s ∈ S ∪ S−1 }.

The corresponding word metric is denoted by dS.

For example, when G = Z and S = {1 }, then the correspondingCayley graph is:

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But which Cayley graph?

However, when G = Z and S = {2,3 }, then the corresponding Cayleygraph is:

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QuestionDoes there exist an Borel choice of generators for each f.g. group suchthat isomorphic groups are assigned isomorphic Cayley graphs?


Standard Borel Spaces

DefinitionA standard Borel space is a Polish space X equipped with itsσ-algebra of Borel subsets.

Some ExamplesR, [0,1], 2N, Gfg , G, ...If σ is a sentence of Lω1ω, then

Mod(σ) = {M = 〈N, · · · 〉 | M � σ}

is a standard Borel space.

Theorem (Kuratowski)There exists a unique uncountable standard Borel space up toisomorphism.


Borel maps

DefinitionLet X , Y be standard Borel spaces.Then the map ϕ : X → Y is Borel iff graph(ϕ) is a Borel subsetof X × Y .Equivalently, ϕ : X → Y is Borel iff ϕ−1(B) is a Borel set for eachBorel set B ⊆ Y .

An Analogue of Church’s ThesisEXPLICIT = BOREL


Stating the main result ...

DefinitionG∗fg is the space of f.g. groups with underlying set N.

TheoremThere does not exist a Borel map ψ : G∗fg → N<N such that forall G, H ∈ G∗fg

ψ(G) generates G.If G ∼= H, then Cay(G, ψ(G)) ∼= Cay(H, ψ(H)).


Equivalently ...

DefinitionLet C be the standard Borel space of graphs Γ with underlying set Nwhich satisfy the following conditions:

Each vertex v ∈ Γ has finite degree.Aut(Γ) acts transitively on Γ.

ObservationC includes the Cayley graphs of f.g. groups.

Proof.If { x , xs } is an edge of the Cayley graph Cay(G,S ) and g ∈ G, then

g · { x , xs } = {gx ,gxs }.


Equivalently ...

TheoremThere does not exist a Borel map ϕ : Gfg → C such that foreach G, H ∈ Gfg

ϕ(G) is a Cayley graph of G.If G ∼= H, then ϕ(G) ∼= ϕ(H).

QuestionWhy is the Theorem “obviously true”?

AnswerBecause the isomorphism problem for f.g. groups is much harderthan that for Cayley graphs.


Borel reductions

DefinitionLet E , F be equivalence relations on the standard Borel spaces X , Y .

E ≤B F iff there exists a Borel map ϕ : X → Y such that

x E y ⇐⇒ ϕ(x) F ϕ(y).

In this case, ϕ is called a Borel reduction from E to F .E ∼B F iff both E ≤B F and F ≤B E .E <B F iff both E ≤B F and E �B F .


Smooth vs Nonsmooth

DefinitionThe equivalence relation E on the standard Borel space X is smoothiff there exists a Borel map ϕ : X → NN such that

x E y ⇐⇒ ϕ(x) = ϕ(y).

Theorem (Folklore)The isomorphism relation on C is smooth.

Theorem (S.T.-Velickovic)The classification problem for f.g. groups is not smooth.


Borel homomorphisms

DefinitionThe Borel map ϕ : ( X ,E )→ ( Y ,F ) is a Borel homomorphism iff

x E y =⇒ ϕ(x) F ϕ(y).

TheoremIf ϕ : 〈 Gfg ,∼= 〉 → 〈 C,∼= 〉 is any Borel homomorphism, then there existgroups G, H ∈ Gfg such that:

ϕ(G) ∼= ϕ(H).G and H don’t have isomorphic Cayley graphs.

QuestionBut how can we be sure that two f.g. groups do not have isomorphicCayley graphs with respect to some finite generating sets?


The basic idea of geometric group theory

Although the Cayley graphs of a f.g. group G with respect to differentgenerating sets S are usually nonisomorphic, they always have thesame large scale geometry.

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The quasi-isometry relation

Definition (Gromov)Let G, H be f.g. groups with word metrics dS, dT respectively. ThenG, H are said to be quasi-isometric, written G ≈QI H, iff there exist

constants λ ≥ 1 and C ≥ 0, anda map ϕ : G→ H

such that for all x , y ∈ G,

1λ

dS(x , y)− C ≤ dT (ϕ(x), ϕ(y)) ≤ λdS(x , y) + C;

and for all z ∈ H,dT (z, ϕ[G]) ≤ C.


When C = 0

Definition (Gromov)Let G, H be f.g. groups with word metrics dS, dT respectively. ThenG, H are said to be Lipschitz equivalent iff there exist

a constant λ ≥ 1, anda map ϕ : G→ H


1λ

dS(x , y) ≤ dT (ϕ(x), ϕ(y)) ≤ λdS(x , y);

and for all z ∈ H,dT (z, ϕ[G]) = 0.


The quasi-isometry relation

Definition (Gromov)Let G, H be f.g. groups with word metrics dS, dT respectively. ThenG, H are said to be quasi-isometric, written G ≈QI H, iff there exist

constants λ ≥ 1 and C ≥ 0, anda map ϕ : G→ H


1λ

dS(x , y)− C ≤ dT (ϕ(x), ϕ(y)) ≤ λdS(x , y) + C;

and for all z ∈ H,dT (z, ϕ[G]) ≤ C.


As expected

ObservationIf S, T are finite generating sets for G, then

id : 〈G,dS〉 → 〈G,dT 〉

is a quasi-isometry.

CorollaryIf the f.g. groups G, H have isomorphic Cayley graphs with respectto some finite generating sets, then G, H are quasi-isometric.


A topological characterization

Theorem (Gromov)If G, H are f.g. groups, then the following are equivalent.

G and H are quasi-isometric.There exists a locally compact space X on which G, H havecommuting proper actions via homeomorphisms such thatX/G and X/H are both compact.

DefinitionThe action of the discrete group G on X is proper iff for every compactsubset K ⊆ X , the set {g ∈ G | g(K ) ∩ K 6= ∅} is finite.


Obviously quasi-isometric groups

DefinitionTwo groups G1, G2 are said to be virtually isomorphic iff there existsubgroups Ni 6 Hi 6 Gi such that:

[G1 : H1], [G2 : H2] <∞.N1, N2 are finite normal subgroups of H1, H2 respectively.H1/N1

∼= H2/N2.

Proposition (Folklore)If the f.g. groups G1, G2 are virtually isomorphic, then G1, G2 arequasi-isometric.


More quasi-isometric groups

Theorem (Erschler)The f.g. groups Alt(5) wr Z and C60 wr Z are quasi-isometric but notvirtually isomorphic. (In fact, they have isomorphic Cayley graphs.)

DefinitionThe f.g. group G is quasi-isometrically rigid iff whenever H isquasi-isometric to G, then H is virtually isomorphic to G.

ExampleZn, Fn, SLn(Z), ... are quasi-isometrically rigid.

QuestionDo there exist uncountably many quasi-isometrically rigid f.g. groups?


Growth rates and quasi-isometric groups

Theorem (Grigorchuk 1984 - Bowditch 1998)

There are 2ℵ0 f.g. groups up to quasi-isometry.

Proof (Grigorchuk).Consider the growth rate of the size of balls of radius n in the Cayleygraphs of suitably chosen groups.

Proof (Bowditch).Consider the growth rate of the length of “taut loops” in the Cayleygraphs of suitably chosen groups.


The complexity of the quasi-isometry relation

QuestionWhat are the possible complete invariants for the quasi-isometryproblem for f.g. groups?

QuestionIs the quasi-isometry problem for f.g. groups strictly harder than theisomorphism problem?


Almost there ...

TheoremIf ϕ : 〈 Gfg ,∼= 〉 → 〈 C,∼= 〉 is any Borel homomorphism, then there existgroups G, H ∈ Gfg such that:

ϕ(G) ∼= ϕ(H).G and H are not quasi-isometric.

RemarkOf course, in order to prove this, we must actually show that there are“many such pairs”.


A little ergodic theory

DefinitionA measure preserving action of a group Γ on a probability space (X , ν)is ergodic iff whenever Y ⊆ X is a Γ-invariant Borel subset,then ν(Y ) = 0, 1.

ExampleLet µ be the usual product probability measure on 2Γ and consider theshift action of Γ on 2Γ = P(Γ). Then µ is Γ-invariant and Γ actsergodically on (2Γ, µ).


A little ergodic theory

TheoremIf Γ y ( X , µ ) is a probability preserving action, then the followingstatements are equivalent.

(i) Γ acts ergodically on (X , µ).(ii) If Y is a standard Borel space and f : X → Y is a Γ-invariant

Borel function, then there exists a Γ-invariant Borel subsetM ⊆ X with µ(M) = 1 such that f � M is a constant function.


Some easy consequences

DefinitionLet EZ be the orbit equivalence relation for the action of Z on 2Z.

PropositionIf ψ : 〈2Z,EZ 〉 → 〈NN,= 〉 is a Borel homomorphism, then there existsa Borel subset M ⊆ 2Z with µ(M) = 1 such that ψ � M is a constantfunction.

CorollaryIf ψ : 〈2Z,EZ 〉 → 〈 C,∼= 〉 is a Borel homomorphism, then there existsa Borel subset M ⊆ 2Z with µ(M) = 1 such that ψ maps M into asingle ∼=-class.


The heart of the matter

Main LemmaThere exists a Borel homomorphism

θ : 〈2Z,EZ 〉 → 〈Gfg ,∼= 〉T 7→ GT

such that the set

{〈S,T 〉 ∈ 2Z × 2Z | GS,GT are not quasi-isometric }

has (µ× µ)-measure 1.


Proof of Theorem

Suppose that ϕ : 〈 Gfg ,∼= 〉 → 〈 C,∼= 〉 is a Borel homomorphism.Consider the composite Borel homomorphism

ψ : 〈2Z,EZ 〉θ−→ 〈Gfg ,∼= 〉

ϕ−→ 〈C,∼= 〉.

Then there exists a Borel subset M ⊆ 2Z with µ(M) = 1 suchthat ψ maps M into a single ∼=-class.Since the set

{〈S,T 〉 ∈ 2Z × 2Z | GS,GT are not quasi-isometric }

has (µ× µ)-measure 1, there exist S, T ∈ M such that GS, GTare not quasi-isometric.


A slight variant of Champetier’s construction

DefinitionLet F3 be the free group on {a,b, c } and let g ∈ Aut(F3) be theautomorphism defined by:

g(a) = ab g(b) = ab2 g(c) = c

LemmaIf w = (ac)14, then the presentation 〈a,b, c | gn(w) 〉n∈Z satisfiesthe C′(1/6) cancellation property.

DefinitionLet θ : 2Z → Gfg be the Borel map defined by

S θ7→ GS = 〈a,b, c | RS 〉, where RS = {gs(w) | s ∈ S }.


θ is a Borel homomorphism

LemmaIf S EZ T , then GS

∼= GT .

Proof.Suppose that T = n + S and consider gn ∈ Aut(F3).

Clearly gn( {gs(w) | s ∈ S } ) = {gt (w) | t ∈ T }.

Hence gn induces an isomorphism from GS = 〈a,b, c | RS 〉onto GT = 〈a,b, c | RT 〉.


Combining ideas of Champetier and Bowditch

LemmaThe “taut loops” in the Cayley graph of GS = 〈a,b, c | RS 〉 correspondprecisely to the relators RS = {gs(w) | s ∈ S }.

LemmaIf GS, GT are quasi-isometric, then

{ length(gs(w)) | s ∈ S } ≈ { length(gt (w)) | t ∈ T }

DefinitionIf D, E ⊆ N+, then D ≈ E , iff there exists k ≥ 1 such that:

For every d ∈ D, there exists e ∈ E such that d/k ≤ e ≤ kd .For every e ∈ E , there exists d ∈ D such that e/k ≤ d ≤ ke.


Probability 101

ObservationIf |`| � 0, then 2|`| 4 length(g`(w)) 4 3|`|.

DefinitionFor each n ≥ 1 and k � 0, let

Dn(k) = {` ∈ Z | |gk (w)|/2n ≤ |g`(w)| ≤ 2n|gk (w)|}.

LemmaThere exists k∗ � 0 such that

Dn(k) ⊆ [ k − 2n, k + 2n ] ∪ [ k∗ − 2n, k∗ + 2n ].


Probability 101

With µ-probability 1, if s ∈ 2Z, then for each n ≥ 1, there existinfinitely many pairwise disjoint sets of the form Dn(k) such thats � Dn(k) ≡ 0.

Fix some such s ∈ 2Z.

With µ-probability 1, if t ∈ 2Z, then for each n ≥ 1, there existinfinitely many pairwise disjoint sets of the form Dn(k) such thats � Dn(k) ≡ 0 and t(k) = 1.

It follows that Gs and Gt are not quasi-isometric.


The quasi-isometry relation ≈QI is not smooth

Suppose that ϕ : Gfg → NN witnesses that ≈QI is smooth.

Let θ : 〈2Z,EZ 〉 → 〈Gfg ,∼= 〉 be a Borel homomorphism such that{〈S,T 〉 ∈ 2Z × 2Z | θ(S) 6≈QI θ(T ) } has (µ× µ)-measure 1.

Then ϕ ◦ θ : 2Z → NN is Z-invariant.

Hence there exists M ⊆ 2Z with µ(M) = 1 such that (ϕ ◦ θ) � Mis a constant map.

But then there exists 〈S,T 〉 ∈ M ×M such that θ(S) 6≈QI θ(T )and ϕ(θ(S)) = ϕ(θ(T )), which is a contradiction.


The complexity of the quasi-isometry relation

ConjectureThe quasi-isometry relation ≈QI for f.g. groups is strictly more complexthan the isomorphism relation ∼=Gfg .

DefinitionThe f.g. groups G1, G2 are virtually isomorphic, written G1 ≈VI G2, ifthere exist subgroups Ni 6 Hi 6 Gi such that:

[G1 : H1], [G2 : H2] <∞.N1, N2 are finite normal subgroups of H1, H2 respectively.H1/N1

∼= H2/N2.

Theorem (S.T.)The virtual isomorphism problem for f.g. groups is strictly harder thanthe isomorphism problem.


Kσ equivalence relations

DefinitionThe equivalence relation E on the Polish space X is Kσ if E is theunion of countably many compact subsets of X × X .

ExampleThe following are Kσ equivalence relations on the space Gfg of finitelygenerated groups:

the isomorphism relation ∼=the virtual isomorphism relation ≈VI

the quasi-isometry relation ≈QI


Some universal Kσ equivalence relations

Theorem (Rosendal)Let EKσ be the equivalence relation on

∏n≥1{1, . . . ,n } defined by

α EKσ β ⇐⇒ ∃N ∀k |α(k)− β(k)| ≤ N.

Then EKσ is a universal Kσ equivalence relation.

Theorem (Rosendal)The Lipschitz equivalence relation on the space of compact separablemetric spaces is Borel bireducible with EKσ .


More universal Kσ equivalence relations

Theorem (S.T.)The following equivalence relations are Borel bireducible with EKσ

the growth rate relation on the space of strictly increasingfunctions f : N→ N;the quasi-isometry relation on the space of connected4-regular graphs.

DefinitionThe strictly increasing functions f , g : N→ N have the samegrowth rate, written f ≡ g, if there exists an integer t ≥ 1 such that

f (n) ≤ g(tn) for all n ≥ 1, andg(n) ≤ f (tn) for all n ≥ 1.


The quasi-isometry vs. virtual isomorphism problems

The Main ConjectureThe quasi-isometry problem for f.g. groups is universal Kσ.

Theorem (S.T.)The virtual isomorphism problem for f.g. groups is not universal Kσ.

CorollaryThe virtual isomorphism problem for f.g. groups is strictly easier thanthe quasi-isometry relation for connected 4-regular graphs.

ConjectureThe virtual isomorphism problem for f.g. groups is strictly easier thanthe quasi-isometry relation for for f.g. groups.

The End


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