2 Means

2.1 The space of means

Definition 2.1 Let (E,N) be a Banach space. Its dual space is the space of all bounded linear maps ER. It is a Banach space with the norm N(λ)=supfE1|λ(f)| where E1 is the unit closed ball of E.

The weak- topology is the coarsest topology on E such that evaluation maps λλ(f) are continuous for all fE. It is a topological vector space (i.e. addition and multiplication maps are continuous maps) that is locally convex: There is a basis of neighborhoods of 0 given by convex open subsets.

Let us recall a crucial fact for us.

Theorem 2.1 (Banach-Alaoglu theorem) Let (E,N) be a Banach space then the unit ball of E is weak- compact.

For a set X, we by (X) the Banach space of all bounded real functions on X with the norm ||f||=supxX|f(x)|.

Definition 2.2 (mean) A mean on X is an element m of (X) such that

  1. For any non-negative bounded function f on X, m(f)0,
  2. m(1X)=1.

We define M(X) to be the set of all means on X.

Any finitely supported probability yields an element of M(X), we denote them by Probf(X). This space is exactly the space of finite combinations of Dirac masses. More generally, any countably supported probability measure on X yields a mean.

Theorem 2.2 We have the following properties

  1. For any mM(X), ||m||=1
  2. The space of means M(X) is a convex weak-* compact subset of (X)
  3. The space Probf(X) is weak-* dense in M(X).

Proof. Let us observe that the first condition implies that if fg then m(f)m(g) since gf0 and m is linear.

  1. For any f(X), f||f||1X and thus m(f)||f||m(1X)=||f||. So ||m||1 and since m(1X)=1, ||m||=1.
  2. The two conditions defining means are convex and closed. Thus the set of means is an infinite intersections of closed convex subsets so it is closed and convex. By the first point it is a part of the unit ball and by Banach Alaoglu theorem, this is a closed subset of a compact Hausdorff space so it is compact.
  3. Assume that Probf(X) is not weak-* dense in M(X) then there is m0M(X)¯Probf(X) , so there is f(X) non-negative such that maxmProbf(X)m(f)<m0(f) (Hahn-Banach theorem). Let μ be this maximum, we have in particular, for m=δx and for xX that f(x)μ<m0(f). So ||f||<m0(f) and we have a contradiction.

To any mean m on X, one can associate a finitely additive probability measure ˆm in the following way:

ˆm(A)=m(1A).

Theorem 2.3 The map mˆm is a bijection between means and finitely additive probability measures.

Proof. Let us prove injectivity first. The space of bounded functions with finite image is dense in (X). So since means are 1-Lipschitz on (X), it suffices to prove that if ^m1=^m2 then m1(f)=m2(f) for all functions f with finite image. Such a function can be written f=ni=1λi1Xi where λiR and (Xi) is a partition of X.

So if m is a mean m(f)=ni=1λiˆm(Xi) and thus if ^m1=^m2 then they coincide on finite image functions.

Let’s prove that the map is surjective. If μ is a finitely additive probability measure, for functions with finite image f=ni=1λi1Xi, we define ¯μ(f)=ni=1λiμ(Xi). With this definition, |¯μ(f)|||f||. Thus ¯μ can be extended by uniform continuity to (X). Moreover, for all non-negative finite image function f, ¯μ(f)0, thus this holds for all non-negative bounded functions. The condition ¯μ(1X)=μ(X)=1 holds by construction.

2.2 Amenable groups

Let G be a group acting on a space X. The group acts on (X) via the following formula:

f(X),xX,gG, (gf)(x)=f(g1x).

In particular, for f=1A, we have:

g1A=1gA.

Actually, g1A(x)=1A(g1x)=1g1xAxgA.

This defines a left action on (X) by linear maps that preserve the norm. Associated to this action, there is an action on the dual (X):

λ(X),f(X),gG, (gλ)(f)=λ(g1f). The action on (X) preserves non-negative functions and constant functions are invariant. So the induced action on (X) let invariant the space of means. Let us see what it does mean for means and finitely additive probability measures.

Let m be a mean, gG and AX,

^gm(A)=gm(1A)=m(g11A)=m(1g1A)=ˆm(g1A)

This means that the action on finitely additive probabilities satisfies the following formula:

gˆm(A)=ˆm(g1A). A mean m is said to be invariant if for any gG, gm=m.

With the correspondence between means and finitely additive probability measures, we have the following alternative definition for amenability of groups.

Definition 2.3 A group G is amenable if there is a invariant mean on G for its action by left multiplications on itself.

Example 2.1 The group Z is amenable.

Proof. Let μ be a cluster point of the sequence μn=12n+1ni=nδi in M(Z). Since for any bounded function f on Z, we have |1μn(f)μn(f)||f(n+1)f(n)|/(2n+1)0, 1μ=μ and it follows that kμ=μ for any kZ.

Let us see a few equivalent definitions of amenability. Below we denote by Prob(G) the set of countably supported probability measures on G (with respect to the σ-algebra of all subsets of G).

Remark. Any element of Prob(G) can be written μ=iNλiδgi where giG, λi0 and iNλi=1. In particular, it can identified with a subset of the unit ball of 1(G). If G is countable then Prob(G) is exactly the set of probability measures on G but if G is not countable then Prob(G) is not the set of all probability measures on G.

Definition 2.4 (Reiter's condition) A group G satisfies Reiter’s condition if for any finite subset EG and ε>0, there is μProb(G) such that ||gμμ||1<ε for any gE.

Definition 2.5 (Følner's condition) A group G satisfies Følner’s condition if for any finite subset EG and ε>0, there is a finite subset F such that for all gE.

|gFΔF|<ε|F|.

Theorem 2.4 Let G be a group. The following are equivalent.

  1. The group G is amenable.
  2. The group G satisfies Reiter’s condition.
  3. The group G satisfies Følner’s condition.

Proof. Let us prove that amenability implies Reiter’s condition. So, assume G is amenable. There is an invariant mean m on G. Since finitely supported probability measures are dense in M(G) for the weak-* topology, we can find a net μiProbf(G) converging to m. This means that for any f(G), μi(f)m(f) (this is the weak-* convergence). For gG, gμi(f)=μi(g1f) and thus gμi converges to gm as well. In particular, gμiμi converges to 0 for the weak topology of 1(G). For a finite set E, consider the space {sEsμμ, μProb(G)}sE1(G).

This is convex subspace whose weak closure contains 0 (The direct sum sE1(G) is endowed with the product topology of the norm topology, its dual is sE(G)) and its weak topology is the product of the weak topologies on each copy of 1(G)). But a convex subspace is weakly closed if and only if it is strongly closed (consequence of Hahn-Banach separation theorem). So the strong closure of the space above also contains 0 and thus, for any ε>0, one can find μProb(G) such that ||sμμ||1<ε for all sE.

Let’s prove that Reiter’s condition implies Følner’s condition. For a finite subset EG and ε>0, let μProb(G) given by Reiter’s condition. For f1(G) and r0, let F(f,r)={tG, f(t)>r} (an upper level subset of f). For non-negative functions f,h1(G) with norm 1, we have |1F(f,r)(t)1F(h,r)(t)|=1 if and only if f(t)rh(t) or h(t)rf(t). Hence for two functions bounded above by 1 we have

|f(t)h(t)|=10|1F(f,r)(t)1F(h,r)(t)|dr. In particular, for sE and μProb(G).

||sμμ||1=tG|sμ(t)μ(t)|=tG10|1F(sμ,r)(t)1F(μ,r)(t)|dr=10tG|1F(sμ,r)(t)1F(μ,r)(t)|dr=10|sF(μ,r)ΔF(μ,r)|dr

So, by the condition satisfied by μ and summing over all elements of E,

10sE|sF(μ,r)ΔF(μ,r)|drε|E|=ε|E|tGμ(t)=ε|E|tGμ(t)01dr=ε|E|10|{tG, μ(t)>r}|dr=ε|E|10|F(μ,r)|dr

In particular, there is an r>0 such that sE|sF(μ,r)ΔF(μ,r)|ε|E||F(μ,r)| and we have the Følner’s condition.

Let’s prove that the Følner’s condition implies amenability. Let E be a finite set and ε>0. We can find an finite subset F=FE,ε (depending on E and ε) such that |gFΔF|ε|F| for any gE. Let us define μF be gFδg|F|Prob(G). We have

||sμFμF||1ε.

Take a cluster point of the net μFE,ε (where finite subsets are ordered by inclusion) in M(G). This yields an invariant mean.

2.3 The growth function of a finitely generated group

Let G be a group generated by a finite set S.The word metric associated to S is the metric dS defined by dS(g,g) is the minimal of a word in S representing g1g. This metric is invariant under left multiplication and it correspond this the length metric in the Cayley graph associated to S.

Definition 2.6 The growth function of G associated to S is the function f:NR given by

f(n)=|BS(e,n)|

where BS(e,n) is the closed ball of radius n around the identity e in the Cayley graph of G associated to S.

Remark. One has f(n)(2|S|+1)n since there are at most kn distinct words of length n on an alphabet with k letters.

Let us see that this growth function is coarsely independent of S.

Definition 2.7 Let (X,d) and (Y,d) be two metric spaces and f:XY. Let (λ,C)R2+. The function f is said to be a (λ,C)-quasi-isometric embedding if for any x,xX,

1/λd(f(x),f(x))Cd(x,x)λd(f(x),f(x))+C. The function f is said to be a quasi-isometric embedding if there exists (λ,C)R2+ such that f is a (λ,C)-quasi-isometric embedding.

The function f is a (λ,C,M)-quasi-isometry if it is a (λ,C)-quasi isometric embedding and for any yY, there is xX such that d(y,f(x))M. Finally, f is said to be a quasi-isometry if there is (λ,C,M)R3+ such that f is a (λ,C,M)-quasi-isometry.

Proposition 2.1 Let G be a group with two finite generating sets S,S. Then the identity map GG is a quasi-isometry between vertices of G(G,S) and G(G,S).

Proof. Let m be the maximum of length of an element of S for the word distance associated to S. So, if g=s1...sn where each element SiS±1 then by replacing each si by a product of elements of S±1 of length at most m, we get that

|g|Sm|g|S. Permuting the role of S and S, we get |g|Sm|g|S for some m and thus the identity is a quasi-isometry.

Definition 2.8 Let G be a group generated by a set S and let f the associated growth function.

  1. G has polynomial growth of degree at most k if f(n)=O(nk).
  2. G has exponential growth if lim supf(n)1/n>1.
  3. G has sub-exponential growth if lim supf(n)1/n=1.
  4. G has intermediate growth if it is neither of polynomial growth or of exponential growth.

Remark. Polynomial growth, exponential growth and sub-exponential growth does not depend on the choice of the finite generating set.

Example 2.2 The groups Zk with its standard generating set (1,0,0),,(0,,0,1) has polynomial growth at most k. The free group Fk has exponential growth with exponent 2k1 for its standard generating set.

Theorem 2.5 A finitely generated group G with sub-exponential growth is amenable.

Proof. Let fix a finite symmetric generating set S for G and let us denote by B(n) the ball of radius n for the word metric on G.

We prove that G satisfies Følner’s condition for E=S (which is sufficient because any finite set can be included in some finite generating set).

We know that lim sup|B(n)|1/n=1 and thus for any ε>0, for all N there is n>N

|B(n+1)|/|B(n)|<1+ε.

Now, |sB(n)B(n)||B(n)||B(n+1)B(n)||B(n)|<ε.

so |sB(n)ΔB(n)||B(n)|<2ε.

Theorem 2.6 Finitely generated abelian groups have polynomial growth

Proof. Let G be a finitely generated abelian group with n generators. The ball of radius kN has at most (2k+1)n elements and thus, it has polynomial growth.

Corollary 2.1 All finitely generated abelian groups are amenable.

Theorem 2.7 Finitely generated nilpotent groups have polynomial growth.

Proof. A group G is nilpotent if there is a sequence of subgroup G0,G1,G2,,Gn such that G0=G, Gi+1=[G,Gi] (which is normal in Gi and Gi+1/Gi is abelian) and Gn={1}. Assume we know that G1=G is finitely generated (see Exercice below). Arguing by induction, it suffices to prove that if G is a finitely generated group with a normal finitely generated subgroup N with polynomial growth and G/N is abelian then G has polynomial growth. Let q1,,qk generators for G/N with preimages g1,,gk in G. Let gG and π(g) its image in G/N, we have π(g)=tα11tαkk since G/N is abelian. So g can be written gα11gαkkn with nN. Let n1,,nm be generators for N. Let S={g1,,gk,n1,,nm}. This is a finite generating set of G. For the word distance associated | |S to this system, |g|S=(|αi|)+|n|S, in particular if the ball BN(r) in N has size bounded by P(r) for some polynomial P then BG(r)(2r+1)nP(r) and thus the growth is polynomial.

Remark. It is not true in general that the commutator subgroup of a finitely generated group is finitely generated.

Corollary 2.2 Finitely generated nilpotent groups are amenable.

2.4 Exercises

Exercise 2.1 Prove that a group G is amenable if and only if there is a right invariant mean on G.

Exercise 2.2 Let X be an uncountable set and A be the σ-algebra generated by countable subsets.

  1. Prove that any element of A is either countable or co-countable (i.e. its complement is countable).
  2. We define m(A)=0 if A is a countable subset of X and m(A)=1 if A is co-countable. Prove that m is a probability measure on A.

Exercise 2.3 Prove that the group of affine transformations of the real line is amenable.

Exercise 2.4 Let G be a group. For an orthogonal representation of G, π:GO(H) where H is a Hilbert space, we say that π almost has invariant vectors if for all ε>0 and FG finite, there is a unitary vector vH such that ||π(g)(v)v||<ε.

The Koopman representation π:GO(2(G)) is given by π(g)(f)(h)=f(g1h) for g,hG and f2(G). Prove that a group G is amenable if and only there is an almost invariant vector in 2(G) for this Koopman representation.

Exercise 2.5 A group G has Kazhdan Property (T), if there is ε>0 such that for any unitary representation π:GU(H) without invariant unit vector, for any unit vector vH, |π(g)(v)v|>ε.

Prove that an amenable group with Kazhdan property (T) is finite. (Hint: use the previous exercise)

Exercise 2.6 Prove that any two regular trees (i.e. trees with finite constant valency at least 3) are quasi-isometric. Deduce that any two free groups on n,m2 generators are quasi-isometric.

Exercise 2.7 Prove that the growth type (bounded, polynomial, sub-exponential, exponential) of a finitely generated group does not depend on the choice of a finite generating system.

Exercise 2.8 Let G be a finitely generated nilpotent group. The goal of this exercise is to prove that its commutator subgroup G=[G,G] is finitely generated. Let S be a finite generating set of G.

  1. Prove that any commutator can be written as a product of iterated commutators [[[s1,s2],s3],,sn] with siS. (Hint: use iteratively the relation xy=yx[x1,y1]).
  2. Prove that there is a finite quantity of such iterated commutators of elements of S.
  3. Conclude that G is finitely generated.

Exercise 2.9 Let G be group. We say that G is virtually something if there is a subgroup HG of finite index such H is something (replace something by some adjective for groups).

  1. Prove that a finite index subgroup H of a finitely generated group G is finitely generated.
  2. Prove that if G has a finite index subgroup then it has a normal finite index subgroup.
  3. Prove that if G is finitely generated and H is a finite index subgroup then there are finite generating sets SG and SH of G and H such that G and H are quasi-isometric for the induced word metrics.
  4. Prove that a finite index subgroup of a finitely generated group of polynomial growth is of polynomial growth as well.
  5. More generally prove that the growth type of two quasi-isometric groups (for some word length associated to two finite generating sets) is the same. We say that the growth is an invariant of quasi-isometry.