# On the ergodic theorem for affine actions on Hilbert space.

IntroductionIn a groundbreaking paper [15], Shalom discovered deep connections between the representation theory of an amenable group and aspects of its large-scale geometry. One motivation for this work, among others, was the development of a "spectral" approach to Gromov's celebrated theorem on the virtual nilpotency of groups of polynomial growth [10] (see also [1, 7, 12, 13, 16]). More precisely, Shalom established, Theorem 1.11 in [15], that if it could be shown that any group of polynomial growth G possessed property [H.sub.FD] (see Definition 2.9), then this would suffice to establish that G would have a finite-index subgroup with infinite abelianization--the key step in Gromov's proof which involves the use of Hilbert's 5th problem. As a means of establishing property [H.sub.FD], Shalom conjectured that for a group of polynomial growth, a sequence of almost fixed points for any affine action with weakly mixing linear part could be obtained by averaging the associated 1-cocycle over an appropriate subsequence of n-balls centered at the identity: see section 6.7 in [15]. This conjecture was partly based on his observation that for such groups a subsequence of the n-balls must possess a strong, quantitative Folner sequence. While Shalom did manage to show that large classes of amenable groups, including polycyclic groups, do have property [H.sub.FD], the proofs are non-geometric, relying in the polycyclic case on deep results of Delorme [6] about the structure of connected, solvable Lie groups. In their paper [3], Cornulier, Tessera, and Valette investigated a generalized version of Shalom's notion of a strong, quantitative Folner sequence, which they called a controlled Folner sequence. Further, they made a significant contribution to Shalom's program through their investigation of averaging properties of groups over controlled Folner sequences, which has directly influenced the approach taken here.

By the results of Cornulier-Tessera-Valette [3] and Tessera [17, 18, 19] many classes of amenable groups are known to possess controlled Folner sequences (see Proposition 1.10 below), these classes roughly corresponding to the classes of groups known to possess property [H.sub.FD]. This motivates the following question:

Question 0.1. Does every finitely generated group admitting a controlled Folner sequence have property [H.sub.FD] of Shalom?

Statement of results

We prove a weak mean ergodic theorem for affine actions of finitely generated amenable groups on Hilbert space. A sequence ([[mu].sub.n]) of regular Borel probability measures on a countable discrete group G forms a Reiter sequence if [parallel][[mu].sub.n] - g x [[mu].sub.n][parallel] [right arrow] 0 for all g [member of] G, where g x [[mu].sub.n](h) = [[mu].sub.n]([g.sup.-1]h). A countable discrete group is said to be amenable if it admits a Reiter sequence.

Theorem A (Weak Mean Ergodic Theorem). Let [pi]: G [right arrow] O(H) be an ergodic orthogonal representation of a finitely generated amenable group G, and let b: G [right arrow] H be a 1-cocycle associated to [pi]. Let S be a finite symmetric generating set for G, and let [absolute value of *] denote the word length in S. If ([[mu].sub.n]) is a Reiter sequence for G, then

[integral] 1/[absolute value of g] b(g) d[[mu].sub.n](g) [right arrow] 0 (0.1)

in the weak topology on H. If [pi] is weakly mixing, then

[integral] 1/[absolute value of g] [absolute value of <b(g), [xi]>] d[[mu].sub.n](g) [right arrow] 0 (0.2)

for all [xi] [member of] H.

Note that while 1/[absolute value of e] is technically undefined, by convention it will be understood to denote 0 throughout.

In the weak mixing case, Theorem A states that the 1-cocycle must be "almost weakly sublinear" in the sense that for any [epsilon] > 0 and [xi] [epsilon] H, the subset consisting of all elements g [member of] G such that [absolute value of <b(g), [xi]>] [greater than or equal to] [epsilon] [absolute value of g] has measure 0 for all left invariant means on G. We show in Theorem 2.4 below that that for a group admitting a controlled Folner sequence (see Definition 1.7), for every "weakly sublinear" 1-cocycle (i.e., one for which for any [epsilon] > 0 and [xi], [member of] H, the subset consisting of all elements g [member of] G such that [absolute value of <b(g), L>] [greater than or equal to] [epsilon] [absolute value of g] is finite) the associated affine action on Hilbert space admits a sequence of almost fixed points. Thus, the obstruction to settling Question 0.1 in the positive is addressing the gap between "measure zero" sets one hand and finite sets on the other. We note, that in the case of a compact representation, almost weak sublinearity is equivalent weak sublinearity. It seems plausible that through additional structural analysis on the 1-cocycle or the group, one may be able to derive weak sublinearity in the general case and close the gap.

Specializing Theorem A to the integers, if b: Z [right arrow] H is a 1-cocycle, then b is completely determined by [xi] [??] b(1), so that for n [greater than or equal to] 1 we have 1/n b(n) = [A.sub.n]([xi]) := 1/n [[summation].sup.n-1.sub.k=0] [pi](k)[xi]: a similar formula holds for -n via the identity b(-n) = -[pi](-n)b(n). So, in this case the result reduces to the fact that the Cesaro sums [C.sub.n]([xi], [eta]) = 1/n [[summation].sup.n.sub.k=1] <[A.sub.k]([xi]), [eta]> and [C'.sub.n]([xi], [eta]) = 1/n [[summation].sup.n.sub.k=1] [absolute value of <[A.sub.k]([xi]), [eta]>] converge to 0 for all [xi], [eta] [member of] H. In fact, the stronger summation holds for all ergodic representations and is equivalent to the (weak) mean ergodic theorem of von Neumann.

In fact, for the class of abelian groups, the above result gives a new, geometrically flavored proof of the mean ergodic theorem in combination with the following result.

Theorem B. Let G be finitely generated amenable group admitting a controlled Folner sequence. Let [pi]: G [right arrow] O(H) bean orthogonal representation, and let b: G [right arrow] H be a 1-cocycle associated to [pi]. Suppose that

[integral] 1/[absolute value of g] <b([g.sup.-1]), [xi]> d[[mu].sub.n](g) [right arrow] 0 (0.3)

for all [xi], [member of] H and all Reiter sequences ([[mu].sub.n]). Then the affine action G [[??].sup.T] H associated to b admits a sequence of almost fixed points.

To see how this implies the mean ergodic theorem for Z, we point out that by an observation of Cornulier-Tessera-Valette (Proposition 3.1 in [3]) a consequence of G [[??].sup.T] H admitting almost fixed points is that

1/[absolute value of g] [parallel]b(g)[parallel] [right arrow] 0

as [absolute value of g] [right arrow] [infinity]; in other words, the 1-cocycle b has sublinear growth. In fact, sublinearity of a 1-cocycle is actually equivalent in general to the mean ergodic theorem, i.e., the statement that

[integral] 1/[absolute value of g] [parallel]b(g)[parallel] d[[mu].sub.n](g) [right arrow] 0

for all Reiter sequences ([[mu].sub.n]) (see Proposition 1.16).

The significance of averaging on the right rather than on the left in Theorem B is that it allows one to conclude that the cocycle is weakly sublinear, i.e., 1/[absolute value of g] b(g) [right arrow] 0 in the weak topology, from which point averaging arguments over a controlled Folner sequence produce the desired sequence of almost fixed points. This key to this argument is the fact that g [??] b(g) is a lipschitz function, i.e., [parallel]b(gs) - b(g)[parallel] is uniformly bounded in g. Alas, this is not necessarily the case for [parallel]b(sg) - b(g)[parallel] which again prevents us from settling Question 0.1.

Remarks on the proofs

The paper is an application of the authors' investigations into the "large scale" properties of affine actions of groups on Hilbert space. Though the above results are stated for affine actions, even in this case the proofs rely on a coarsening of the notion of an affine action, the concept of an array, formalized by the authors in [2]. The main novelty of this viewpoint is that it allows one to construct the "absolute value" of a 1-cocycle b: G [right arrow] H which lies in the G-invariant positive cone V [subset] H [cross product] H which allows one to naturally use the weak mixingness to derive the stronger ergodic theorem in that case. Note that such a map cannot lie within a uniformly bounded distance of an (unbounded) 1-cocycle, since the equation b(g) = -[[pi].sub.g]b([g.sup.-1]) holds for all g [member of] G for any 1-cocycle b.

The notion of an array is best viewed from a geometric, rather than algebraic, perspective. Indeed, a length function on a discrete group G may be viewed as a positive array associated with the trivial representation. In general, an array can be thought of as a Hilbert-space valued length function on G which is compatible with some orthogonal G-representation n. The presence of an array then becomes a tool through which properties of the representation can be used to impose large scale conditions on the group, and vice versa. For example, it is shown in [2], Proposition 1.7.3, that a non-amenable group admitting a proper array into its left-regular representation, e.g., non-elementary Gromov hyperbolic groups, cannot be decomposed as a direct product of infinite groups. Turning to the topic at hand, the presence of a controlled Folner sequence imposes a strong large-scale "finite dimensionality" condition on the group G--for the case of weak polynomial growth, a point already well made in [10]. Viewed in this light, the content of Theorem B is that this forces any geometric realization of the group which is uniformly distributed throughout an infinite-dimensional Hilbert space to be essentially degenerate.

1 Geometry and Representation Theory

In this section we will introduce the main definitions and concepts used in the sequel.

Notation 1.1. Let X be a set and let f, g: X [right arrow] [R.sub.[greater than or equal to]0] be maps. We write f [much less than] g if there exists a finite set F [subset] X and a constant C > 0 such that f(x) [less than or equal to] C x g(x) for all x [member of] X\F. We will write f [??] g if f [much less than] g for a constant C [less than or equal to] 1.

1.1 Isometric actions on Hilbert space

Definition 1.2. An orthogonal representation n: G [right arrow] O(H) is said to be ergodic if for any [xi] [member of] H we have that [[pi].sub.g]([xi]) = [xi] for all g [member of] G if and only if [xi] = 0, i.e., [pi] has no non-zero invariant vectors. The representation [pi] is said to be weakly mixing if the diagonal representation [pi] [cross products] n: G [right arrow] O(H [cross product] H) is ergodic. In particular weakly mixing representations are ergodic.

If [pi]: G [right arrow] O(H) is an orthogonal representation, a map b: G [right arrow] H is said to be a 1-cocycle associated to [pi] if it satisfies the Leibniz identity

b(gh) = [[pi].sub.g](b(h)) + b(g),

for all g, h [member of] G. It is essentially a consequence of the Mazur-Ulam theorem that any isometric action G [[??].sup.T] H may be written as [T.sub.g]([xi]) = [[pi].sub.g] ([xi]) + b(g) for some orthogonal representation [pi] and an associated 1-cocycle b(g) and conversely. The representation n is known as the linear part of T.

Definition 1.3. An isometric action G [[??].sup.T] H is said to admit almost fixed points if there exists a sequence ([[xi].sub.n]) of vectors in H such that

[parallel][T.sub.g] ([[xi].sub.n]) - [[xi].sub.n][parallel] [right arrow] 0

for all g [member of] G.

Definition 1.4. We will say that a 1-cocycle b associated to an orthogonal representation [pi]: G [right arrow] O(H) is almost inner if the associated affine isometric action G [??] H admits almost fixed points.

1.2 Geometric group theory

Throughout the paper G will be a countable discrete group, often finitely generated. Recall that a length function [absolute value of *]: G [right arrow] [R.sub.[greater than or equal to]0] is a map satisfying: (1) [absolute value of g] = 0 if and only if g = e is the identity; (2) [absolute value of [g.sup.-1]] = [absolute value of g], for all g [member of] G; and (3) [absolute value of gh] [less than or equal to] [absolute value of g] + [absolute value of h], for all g, h [member of] G. A length function is proper if the map g [??] [absolute value of g] is proper, i.e., all sets of bounded length are finite. If [absolute value of *] is a length function, then we denote

B(n) = {g [member of] G: [absolute value of g] [less than or equal to] n},

the ball of radius n centered at the identity, and

S(n) = {g [member of] G: [absolute value of g] = n},

the sphere of radius n centered at the identity. If G is generated by a finite set S, then the function which assigns to each g [member of] G the least integer k such that g can be written as a product of k elements from S [union] [S.sup.-1] is a proper length function, known as a word length function.

Notation 1.5. Let G be a finitely-generated discrete group with a fixed finite, symmetric, generating set S. Let F [subset] G be a finite subset. We set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where "[DELTA]" denotes the symmetric difference.

Definition 1.6. A sequence [([F.sub.n]).sub.n[member of]N] of finite subsets of G is said to form a Folner sequence if

[absolute value of g[F.sub.n][DELTA][F.sub.n]]/[absolute value of [F.sub.n]] [right arrow] 0

for all g [member of] G.

Definition 1.7. Let G be a finitely generated discrete group with a fixed finite, symmetric, generating set S. For a constant K > 0, a sequence [([F.sub.n]).sub.n[member of]N] of finite subsets of G is said to be a K-controlled Folner sequence if

[absolute value of [partial derivative][F.sub.n]]/[absolute value of [F.sub.n]] [less than or equal to] K/diam [F.sub.n],

where diam [F.sub.n] is defined to be the least integer m such that [F.sub.n] [subset] B(m). The group admits a controlled Folner sequence if it admits a K-controlled Folner sequence for some K.

Definition 1.8. A finitely generated group G is said to have polynomial growth if for some (equivalently, for any) proper word length function we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The group G is said to be of weak polynomial growth if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any proper word length.

The following observation is due to Shalom.

Proposition 1.9 (Shalom, Lemma 6.7.3 in [15]). If G is a finitely generated group of polynomial growth of degree d, then for any proper word length, there is a subsequence S [subset] N such that the sequence of balls [(B(n)).sub.n[member of]S] form a K-controlled Folner sequence for K > 10d.

In fact, a group G which satisfies a doubling condition [absolute value of B (2n)] [less than or equal to] C x [absolute value of B (n)] for some subsequence admits a controlled Folner sequence by an observation of Tessera, [18], Remark 4.10. Gromov's "Regularity lemma" ([10], section 3) shows that groups of weak polynomial growth have the doubling condition. By the work of Tessera several large classes of groups are known to admit controlled Folner sequences.

Proposition 1.10 (Tessera, Theorem 11 in [18] and Theorem 6 in [19]). The following classes of groups admit controlled Folner sequences:

1. polycyclic groups;

2. wreath products D [??] Z with D finite;

3. semi-direct products Z[1/mn] [x.sub.m/n] Z, with m, n coprime and [absolute value of mn] [greater than or equal to] 2;

4. any closed, undistorted subgroup (e.g., cocompact lattice) of a direct product of a p-adic solvable group with a connected, solvable Lie group.

By results of Mal'cev and Auslander, it is known that a group G is polycyclic if and only if G is realizable as a solvable subgroup of GL(n, Z), cf. [5], section III.A.5.

The full extent of the class of amenable groups admitting a controlled Folner sequence is unknown. An interesting problem would be to determine exactly which solvable groups with finite Hirsch number belong to this class or at least have property [H.sub.FD]. (To recall, let G be a solvable group with derived series G > [G.sup.(1)] > [G.sup.(2)] > ... > [G.sup.(n)] > [G.sup.(n+1)] = {1}. The Hirsch number is then defined to be the sum of the torsion-free ranks of the abelian groups [G.sup.(i)]/[G.sup.(i+1)], i = 1, ..., n. See section 6.6 in [15] for a discussion on this problem.) We pose the following, more concrete question:

Question 1.11. If F is a solvable subgroup of GL (n, Z[1/p]), does [GAMMA] admit a controlled Folner sequence?

If [GAMMA] is an undistorted solvable subgroup of GL(n, Z[1/p]), then the answer is affirmative by item (4) of the previous proposition, so it would be interesting to know whether there are other solvable subgroups of GL(n, Z[1/p]). We remark that Z[1/p] cannot be replaced with Z[[tau]] for some non-algebraic number [tau], since GL(2, Z[[tau]]) contains a copy of Z [??] Z which does not to admit a controlled Folner sequence by an isoperimetric inequality due to Erschler [8].

1.3 Arrays

The definition of an array was formally introduced in [2] as a means for unifying the concepts of length functions and 1-cocycles into orthogonal representations. We now recall the definition.

Definition 1.12. Let [pi]: G [right arrow] O(H) be an orthogonal representation of a countable discrete group G. A map [alpha]: G [right arrow] H is called an array if for every finite subset F [subset] G there exists K [greater than or equal to] 0 such that

[parallel][[pi].sub.g]([alpha](h)) - [alpha](gh)[parallel] [less than or equal to] K, (1.1)

for all g [member of] F, h [member of] G (i.e., [alpha] is boundedly equivariant). It is an easy exercise to show that for any array [alpha] on a finitely generated group G there exists a proper word length function on G, a scalar multiple of which bounds [parallel][alpha](g)[parallel] from above.

Lemma 1.13. Let G be a finitely generated group equipped with some proper word length associated to a finite, symmetric, generating set S. If [alpha]: G [right arrow] O(H) is an array into an orthogonal representation [pi], then [??](g) := 1/[absolute value of g] [alpha](g) [cross product] [alpha](g), with [??](e) := 0, is an array into [pi] [cross product] [pi].

Proof. The proof is very similar to the proof of Proposition 1.4 of [2]: we include it here only for the sake of completeness. First, for every g [member of] G, we denote by [B.sub.g] := [sup.sub.h[member of]G] [parallel][alpha](gh) - [[pi].sub.g]([alpha](h))[parallel] and from the assumptions we have [B.sub.g] < [infinity]. Using the triangle inequality together with the bounded equivariance property, for all k [member of] G we have [parallel][alpha](k)[parallel] [less than or equal to] D [absolute value of k], where D = [max.sub.s[member of]S] [B.sub.s]. This further implies that for every l [member of] G we have the following inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.2)

To check the bounded equivariance for a, we fix g, h [member of] G where h [not equal to] e, [g.sup.-1]. Applying the triangle inequality and using successively the bounded equivariance property, the basic inequality [parallel]gh| - |h[parallel] [less than or equal to] [absolute value of g], and the inequality (1.2), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This implies that for every g, h [member of] G we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which concludes our proof as the right hand expression depends only on g.

1.4 Large scale lipschitz maps

Let V be a normed vector space. We will say a map f: G [right arrow] V is large scale lipschitz if there exists a map C: G [right arrow] [R.sub.[greater than or equal to]0] such that for all g [member of] G, [parallel]f(g) - f(gs)[parallel] [less than or equal to] C(s). An array can be viewed in some sense as the formal adjoint of some large scale lipschitz map f: G [right arrow] H with respect to the representation [pi], viz.,

Proposition 1.14. If [alpha]: G [right arrow] H is an array associated to [pi], then [[alpha].sup.*](g) := [pi](g)[alpha]([g.sup.-1]) is large scale lipschitz. Conversely, if f: G [right arrow] H is large scale lipschitz, then [f.sup.*](g) := [pi](g)f([g.sup.-1]) is an array associated to n.

The proof consists of a straightforward check that the respective identities are satisfied.

Given a finite, symmetric generating set S for G, for any map f: G [right arrow] R we define the variation function [partial derivative]f: G [right arrow] [R.sup.S] by [partial derivative]f(g)(s) := f(g) - f(gs).

Definition 1.15. A bounded function f: G [right arrow] R is said to be slowly oscillating if [parallel][partial derivative]f[parallel] [member of] [C.sub.0] (G), where [parallel]*[parallel] is the euclidean norm on [R.sup.S].

Note that if f: G [right arrow] V is a large scale lipschitz map into a normed vector space V, then g [??] 1/[absolute value of g] f(g) is slowly oscillating.

We define [H.sup.[infinity]](G) to be Banach space of all slowly oscillating functions. For all 1 [less than or equal to] p < [infinity], we also define [H.sup.p](G) to be the Banach space of all slowly oscillating functions f such that [parallel][partial derivative]f[parallel] [member of] [l.sup.p](G). Note that the definition of [H.sup.p](G) for all 1 [less than or equal to] p [less than or equal to] [infinity] does not depend on the choice of finite generating set.

Our interest in slowly oscillating functions stems from the following "rigidity" phenomenon which can be observed under the assumption of ergodicity.

Proposition 1.16. If f [member of] [H.sup.[infinity]](G) is a function such that

[integral] f([g.sup.-1])d[[mu].sub.n](g) [right arrow] 0

for all Reiter sequences ([[mu].sub.n]), then f [member of] [C.sub.0](G).

Proof. Suppose by contradiction that f does not belong to [C.sub.0](G). Without loss of generality, we would have that there would exist c > 0 and a sequence ([g.sub.n]) of elements in G such that f([g.sub.n]) [greater than or equal to] c for all n [member of] N. Since f [member of] [H.sup.[infinity]](G), for any finite subset F [subset] G there exists n [member of] N sufficiently large so that f(h) [greater than or equal to] c/2 for all h [member of] [g.sub.n]F. Hence, passing to a subsequence of ([g.sub.n]), there is a Folner sequence ([F.sub.k]) with the property that f(h) [greater than or equal to] c/2 for all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all k [member of] N. Taking [[mu].sub.k] to be the uniform probability measure on the set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we would then have constructed a Reiter sequence such that lim [inf.sub.k] [integral] f([g.sup.-1])d[[mu].sub.k](g) [greater than or equal to] c/2 > 0, a contradiction.

Definition 1.17. Let G be an amenable group, and let f: G [right arrow] V be a large scale lipschitz map. We say that f has sublinear growth if lim [sup.sub.[absolute value of g][greater than or equal to]n] [parallel]f(g)[parallel]/[absolute value of g] = 0. We say that f has almost sublinear growth if [integral] 1/[absolute value of g] [parallel]f(g)[parallel]d[[mu].sub.n](g) [right arrow] 0 for all Reiter sequences ([[mu].sub.n]).

Proposition 1.18. Let G be an amenable group. Let f: G [right arrow] H be a large scale lipschitz map in to Hilbert space. If f is symmetric, i.e., [parallel]f(g)[parallel] = [parallel]f([g.sup.-1])[parallel], then the following statements are equivalent:

1. f has sublinear growth;

2. f has almost sublinear growth;

3. [f.sub.[xi]](g) := <f(g), [xi]> has sublinear growth for all [xi] [member of] H and the set V := [{1/[absolute value of g] f(g)}.sub.g[member of]G] is precompact;

4. [f.sub.[xi]] has almost sublinear growth for all [xi] [member of] H and V is precompact.

Proof. The implications (1)[??](2), (1)[??](3), and (3)[??](4) are trivial, while the implication (2)[??](1) follows directly by Proposition 1.16 applied to the function 1/[absolute value of g] [parallel]f(g)[parallel]. Therefore, we only need prove the implication (4)[??](1).

To this end, note that if V is precompact, then for any [epsilon] > 0 we can find a set of vectors [[xi].sub.1], ..., [[xi].sub.n] [member of] H so that

[integral] [(1/[absolute value of g] [parallel]f(g)[parallel]).sup.2] d[mu](g) [less than or equal to] C [n.summation over (i=1)] [integral] 1/[absolute value of g] [absolute value of f(g), [[xi].sub.i]>] d[mu](g) + [epsilon] (1.3)

holds for any probability measure [mu], where C := [sup.sub.g[member of]G] [parallel]f(g)[parallel]/[absolute value of g] < [infinity]. Thus, by almost sublinear growth of each [f.sub.[xi]] and the Cauchy-Schwarz inequality, we have that [integral] 1/[absolute value of g] [parallel]f(g)[parallel] d[[mu].sub.n](g) [right arrow] 0 along any Reiter sequence. By symmetry, the result then obtains by Proposition 1.16.

2 Main Results

2.1 Arrays and the weak mean ergodic theorem

In this section we present the proof of Theorem A. Though the theorem was stated explicitly for cocycles, the natural context for the theorem is actually the class of arrays. This is essentially due to the fact that there is no well-defined product of cocycles, while such a product exists for the class of arrays. This allows us to exploit the weak mixingness in order to derive the strong form of the theorem in that case.

Theorem 2.1 (Theorem A). Let [pi]: G [right arrow] O(H) bean ergodic orthogonal representation of a finitely generated amenable group G, and let [alpha]: G [right arrow] H bean array. Let S be a finite, symmetric, generating set for G, and let [absolute value of *] denote the word length in S. If [([[mu].sub.n]).sub.u[member of]N] is a Reiter sequence for G, then

[integral] 1/[absolute value of g] [alpha](g) d[[mu].sub.n](g) [right arrow] 0 (2.1)

in the weak topology. If [pi] is weakly mixing, then

[integral] 1/[absolute value of g] [absolute value of <[alpha](g), [xi]>] d[[mu].sub.n] (g) [right arrow] 0 (2.2)

for all [xi] [member of] H.

Before we begin the proof, we pause to introduce some convenient notation to be used here as well as in the sequel.

Notation 2.2. Let [alpha]: G [right arrow] H be an array. We set

[[alpha].sup.[flat]](g) = 1/[absolute value of g] [alpha](g),

where by convention [[alpha].sup.[flat]](e) = 0. H [cross product] H will be denoted as [??]. The representation [pounds sterling] [cross product] [pi]: G [right arrow] O([??]) will be denoted as [pi]. The array [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is defined as

[??](g) = 1/[absolute value of g] [alpha](g) [cross product] [alpha](g),

where [??](e) = 0 by convention.

Proof of Theorem 2.1. The proofs of these formulas are inspired by the standard approach to the (weak) mean ergodic theorem for amenable groups. We begin by proving (2.1). To this end, we fix [epsilon] > 0, n [member of] N and note that there exists a finite subset [F.sub.n] [subset] G such that

[parallel][[alpha].sup.[flat]](gh) - [pi](g)[[alpha].sup.[flat]](h)[parallel] [less than or equal to] [epsilon]

whenever g [member of] B (n) and h [member of] G\Fn. Let [xi] [member of] H be a vector of the form [xi] = (1 - [pi]([g.sup.-1]))[eta] for some g [member of] B(n), [eta] [member of] H. We then have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.3)

since [lim.sub.N] [[mu].sub.n] ([F.sub.n]) = 0 and [parallel][[alpha].sup.[flat]][parallel] is bounded. By inspection, the estimate holds for the span V := span{[xi]: [there exist]g [member of] G, [eta] [member of] H([xi] = (1 - [pi](g))[eta])}, establishing the theorem in that case. Since [integral] [parallel][[alpha].sup.[flat]](g)[parallel] d[[upsilon].sub.n](g) is uniformly bounded, the result then extends to the closure of V, which by ergodicity is all of H. This concludes the proof of (2.1).

For the proof of the second part, formula (2.2), we note that if [alpha]: G [right arrow] H is an array for [pi], then [??](g) is an array for [??] by Lemma 1.13. Applying this, we see that

[absolute value of [integral] <[[??].sup.[flat]](h), [xi] [cross product] [xi]> d[[mu].sub.N](h)] = [integral] [[absolute value of <[[alpha].sup.[flat]](h), [xi]>].sup.2]d[[mu].sub.N](h) [right arrow] 0 (2.4)

by the proof of (2.1). By the Cauchy-Schwarz inequality, we have that

[integral] [absolute value of <[[alpha].sup.[flat]](h), [xi]>] d[[mu].sub.N](h) [less than or equal to] [([integral][[absolute value of <[[alpha].sup.[flat]](h), [zeta]].sup.2] d[[mu].sub.N](h)).sup.1/2], (2.5)

and we are done.

In the case the 1-cocycle is proper, there is a sharpening of the above result. The proof is identical the the proof of the previous theorem, using Proposition 1.4 from [2] instead of Lemma 1.13.

Proposition 2.3. Let [pi]: G [right arrow] H be a weakly mixing orthogonal representation. If b: G [right arrow] H is a proper 1-cocycle, then

[integral] 1/[parallel]b(g)[parallel] [absolute value of <b(g), [xi]>] d[[mu].sub.n](g) [right arrow] 0 (2.6)

for all Reiter sequences ([[mu].sub.n]).

2.2 Theorem B and the mean ergodic theorem

We begin with the main technical theorem in this section, the formulation and proof of which are inspired by Lemma 3.4 in [3].

Theorem 2.4. Let G be a finitely generated discrete group in the class CF. Let b: G [right arrow] H be a 1-cocycle associated to an orthogonal representation n. Assume that

1/[absolute value of g] <b(g), [xi]> [member of] [C.sub.0](G) (2.7)

for all [xi] [member of] H (i.e., b is weakly sublinear). Let [([F.sub.n]).sub.n[member of]N] be a K-controlled Folner sequence. Let [[upsilon].sub.n] be the uniform measure on [F.sub.n]. There exists a sequence ([[mu].sub.k]) of finitely supported measures which are in the convex hull of {[[mu].sub.n]} such that [[xi].sub.k] := [integral] b(g) d[[mu].sub.k](g) form a sequence of almost fixed points for the affine action G [[??].sup.T] H associated to b.

Proof. Fix a word length [absolute value of *] coming from some finite, symmetric generating set S [subset] G. Let [d.sub.n] = diam [F.sub.n]. We set [F.sub.n](g) = g[F.sub.n][DELTA][F.sub.n] [subset] [partial derivative][F.sub.n] [subset] B([d.sub.n] + 1), for each g [member of] S. Let [[eta].sub.n] = [integral] b(g) d[[upsilon].sub.n](g).

For all n [member of] N we have the a priori estimate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.8)

where C = [sup.sub.s[member of]S] [parallel]b(s)[parallel].

Therefore, we need only show that for any [xi] [member of] H and g [member of] S, we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.9)

Indeed, the sequence [([T.sub.g]([[eta].sub.n]) - [[eta].sub.n]).sub.n[member of]N] would then have 0 as a weak limit point for any g [member of] S. Thus, the sequence [[direct sum].sub.g[member of]S]([T.sub.g]([[eta].sub.n]) - [[eta].sub.n]) [subset] [[direct sum].sub.g[member of]S] H converges weakly to 0, so that by passing to the convex hull, the theorem obtains.

We now fix [xi] [member of] H. By assumption 2.7 for every [epsilon] > 0 there exists a finite set [E.sub.[epsilon]] [subset] G such that

[absolute value of <b(g), [xi]>] < [epsilon][absolute value of g] (2.10)

for all g [member of] G\[E.sub.[epsilon]].

Since [lim.sub.n][[upsilon].sub.n]([E.sub.[epsilon]]) = 0, we have that for any g [member of] S,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.11)

and we are done.

Question 2.5. By Proposition 3.1 in [3] we know that any almost inner 1-cocycle has sublinear growth. For a general amenable group, is it the case that any weakly sublinear 1-cocycle is in fact (strongly) sublinear?

Examining the proof of the previous theorem, we find that the conclusion holds under the following weaker hypothesis.

Proposition 2.6. Let G be a finitely generated discrete group in the class CF. Let b: G [right arrow] H be a 1-cocycle associated to an orthogonal representation n. For every c > 0, [xi] [member of] H define the set [E.sub.c]([xi]) := {g [member of] G: [absolute value of <b(g), [epsilon]>] [greater than or equal to] c[absolute value of g]}. Suppose there exists K and [([F.sub.n]).sub.n[member of]N] a K-controlled Folner sequence so that for all c, [epsilon], [delta] > 0 [absolute value of [partial derivative][F.sub.n] [intersection] [E.sub.c]([xi])] [less than or equal to] [delta]/[d.sub.n] x [absolute value of [F.sub.n]] for all n sufficiently large. Let [[upsilon].sub.n] be the uniform measure on [F.sub.n]. There exists a sequence ([[mu].sub.k]) of finitely supported measures which are in the convex hull of {[[upsilon].sub.n]} such that [[epsilon].sub.k] := [integral] b(g) d[[mu].sub.k](g) form a sequence of almost fixed points for the affine action G [[??].sup.T] H associated to b.

Proof. The proof of Theorem 2.4 carries over nearly identically, except for the last estimate of equation 2.11. Using the same notation and set-up, fixing [epsilon] > 0 we have instead that for n sufficiently large

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Question 2.7. Suppose G admits a controlled Folner sequence and that E [subset] G be a set which has zero measure for any left invariant mean on G. Does G also admit a controlled Folner sequence ([F.sub.n]) so that for every [delta] > 0 [absolute value of [partial derivative][F.sub.n] [intersection] E] [less than or equal to] [delta]/[d.sub.n] x [absolute value of [F.sub.n]] for all n sufficiently large.

Remark 2.8. Recently, Gournay [9] generalized the argument of Proposition 3.1 in [3] from groups with controlled Folner sequences to the more general class of "transport amenable" groups; see Definition 1.3 in [9]. This class includes, in particular Z [??] Z. Therefore, it would be highly interesting to know whether Theorem 2.4 likewise holds for all transport amenable groups.

Definition 2.9. A group G has property [H.sub.FD] of Shalom if any affine action G [[??].sup.T] H on Hilbert space with weakly mixing linear part admits almost fixed points.

Proposition 2.10. Suppose that either Question 2.5 or 2.7 has a positive solution for a group G which admits a controlled Folner sequence. Then G has property [H.sub.FD].

The proof is an easy consequence of Theorem 2.4 and Proposition 2.6. It follows from an argument given in [15] (Theorem 6.7.2) that a positive solution to either Question 2.5 or 2.7 for all groups of polynomial growth implies Gromov's theorem.

A stated in Proposition 1.10, among the known classes of amenable groups which admit controlled Folner sequences are: groups of (weak) polynomial growth; polycyclic groups, i.e., lattices in solvable Lie groups; wreath products D [??] Z with D finite; semi-direct products Z[1/mn] [x.sub.m/n] Z, with m, n coprime and [absolute value of mn] [greater than or equal to] 2. The latter three classes are the work of Tessera, Theorem 11 in [18]. Each of these classes is known have property [H.sub.FD] by the seminal work of Shalom, Theorems 1.13 and 1.14 in [15], which in the polycyclic case relies in turn on deep work of Delorme [6]from the 1970s. The advantage to the approach suggested here is that it may potentially offer a broad, conceptually unified way of deriving property [H.sub.FD] for large classes of groups.

We also point out that another consequence of a positive solution to Question 2.14 would give an alternate proof of the fact (due to Erchler [8]) that Z [??] Z, for instance, does not admit a controlled Folner sequences, cf. Theorem 1.15 in [15].

Theorem 2.11 (Theorem B). Let G be finitely generated group in the class CF. Let [pi]: G [right arrow] O(H) be an orthogonal representation, and let b: G [right arrow] H be a 1-cocycle associated to [pi]. Suppose that

[integral] 1/[absolute value of g] <b([g.sup.-1]), [xi]>d[[mu].sub.n](g) [right arrow] 0 (2.12)

for all [xi] [member of] H and all Reiter sequences ([[mu].sub.n]). Then the affine action G [[??].sup.T] H associated to b admits a sequence of almost fixed points.

Proof. The proof follows directly from Proposition 1.16 combined with Theorem 2.4.

Definition 2.12. Let G be a finitely generated group and let p be a probability measure on G. A function u: G [right arrow] V into a vector space is said to be [mu]-harmonic if

u(g) = [integral] u(gs)d[mu](s) (2.13)

for all g [member of] G.

Let p be a probability measure with finite second moment, i.e., [integral] [[absolute value of g].sup.2]d[mu](g) < [infinity]. We know, by Theorem 6.1 in [14] and Theorem 6.1 in [4], that every group G without property (T) of Kazhdan admits at least one [mu]-harmonic 1-cocycle for some (irreducible) representation.

Proposition 2.13. Let G be a group in the class CF, [pi]: G [right arrow] O(H) be an orthogonal representation, and b: G [right arrow] H be a [mu]-harmonic 1-cocycle with p having finite second moment. Let [[pi].sub.0] be the restriction of [pi] to the (invariant) subspace [H.sub.0] spanned by the image of b. If V := {1/[absolute value of g] b(g)} is precompact, then [[pi].sub.0] is compact.

Proof. Suppose by contradiction that [H.sub.0] contains an non-zero invariant subspace K on which the restriction of n is weakly mixing. Setting b': G [right arrow] K defined by b'(g) := [P.sub.K]b(g), we then would have that b' is a harmonic 1-cocycle into a weakly mixing representation such that V' := [P.sub.K]V = {1/[absolute value of g] b'(g)} is precompact. Proposition 1.18 then implies that b'(g) has sublinear growth; hence, by Theorem 2.4 it is almost inner. However, no non-zero harmonic 1-cocycle into an orthogonal representation can be almost inner, cf. Theorem 6.1 in [4]. Therefore, b' [equivalent to] 0 which contradicts the fact that the span of V' is dense in K. Thus, we have shown that [[pi].sub.0] contains no non-zero, weakly mixing subrepresentation which implies that [[pi].sub.0] is compact.

Question 2.14. Let G be an amenable group, and let [mu] be a probability measure with finite second moment and trivial Poisson boundary. If u: G [right arrow] R is a lipschitz [mu]-harmonic function such that

[integral] 1/[absolute value of g] [absolute value of u(g)] d[[mu].sub.n](g) [right arrow] 0

for all Reiter sequences ([[mu].sub.n]), does u have sublinear growth?

Notice that if u is harmonic, then [absolute value of u] is subharmonic, i.e., [absolute value of u] (g) [less than or equal to] 1/[absolute value of S] [[summation].sub.s[member of]S] [absolute value of u](gs) for all g [member of] G, so the conjecture may be posed in this generality. A positive solution to Question 2.14 also implies that the group G has property [H.sub.FD]. The use of harmonicity as a tool for "regularizing" the cocycle is a key insight in the approach of Kleiner [13].

Remark 2.15. A result of Hebisch and Saloff-Coste, Theorem 6.1 in [11], shows that there a no non-constant real-valued harmonic functions of sublinear growth on a group of polynomial growth. It would be interesting if a variant of this argument could be made to apply to harmonic functions of almost sublinear growth.

2.3 On the space [H.sup.P](G)

As a last remark, we develop another line of thought towards establishing the mean ergodic theorem for affine actions of groups of polynomial growth independently of Gromov's theorem.

Theorem 2.16. Let G be a one-ended group with a finite, symmetric, generating set S. If f [member of] [H.sup.1](G), then f [member of] [C.sub.0](G) + C1.

Proof. For every [epsilon] > 0, choose r sufficiently large so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since G is one-ended G\[B.sub.r] contains exactly one infinite connected component [U.sub.r]. For every pair of elements g, h [member of] [U.sub.r] there exists a sequence of elements [x.sub.1], ..., [x.sub.n] in [U.sub.r] such that g = [x.sub.1], h = [x.sub.n] and [x.sup.-1.sub.i+1] [x.sub.i] [member of] S for all i = 1, ..., n - 1. Hence it follows by the triangle inequality that

[absolute value of f(g) - f(h)] [less than or equal to] [K.sub.r]

which proves the claim.

In fact, in the case that f is positive, a slightly weaker condition will suffice:

Theorem 2.17. For f [member of] [l.sup.[infinity]](G) and F [member of] [l.sup.[infinity]](G x S), let f x F(g, s) := f(g)F(g, s). Let G be a one-ended group with a finite, symmetric, generating set S. Suppose that f [member of] [l.sup.[infinity]](G), f [greater than or equal to] 0. If [parallel]f x [partial derivative]f[parallel] [member of] [l.sup.1](G), then f [member of] [C.sub.0](G) + Cl.

Note that since f [greater than or equal to] 0, we have that [parallel]f x [partial derivative]f[parallel] [less than or equal to] [parallel][partial derivative]([f.sup.2])[parallel]; hence, by the boundedness of f and standard estimation techniques it follows that if [f.sup.P] [member of] [H.sup.1](G) for any 1 [less than or equal to] p < [infinity], then it holds that f [member of] [C.sub.0](G) + C1.

Proof. Let [GAMMA] = [GAMMA](G, S) be the Cayley graph of G with respect to the generating set S. We produce a new graph [GAMMA]' by subdividing each edge in [GAMMA] so the the vertex set of [GAMMA]' may be identified with V([GAMMA]) [??] E([GAMMA]) and [GAMMA]' is again one-ended. We define a map f': V([GAMMA]') [right arrow] R by f'(g) := f[(g).sup.2] for g [member of] V([GAMMA]) and f'(e) := f(g)f(gs) for e = (g, gs) [member of] E([GAMMA]). Now by assumptions we can see that [parallel][partial derivative]f'[parallel] [member of] [l.sup.1](V([GAMMA]')), so by Theorem 2.16, we can conclude that [f.sup.2] [member of] [C.sub.0](G) + C1. By the positivity of f, this suffices to show the result.

Proposition 2.18. Let G be a one-ended group in the class CF. If b is a 1-cocycle associated to an ergodic representation [pi]: G [right arrow] O(H) such that

1/[absolute value of g] <b(g), [xi]> [member of] [H.sup.1] (G) (2.14)

for all [xi] [member of] H, then b is almost inner. The same holds assuming that [pi] is weakly mixing and

1/[absolute value of g] [absolute value of <b(g), [xi]>] [member of] [H.sup.1](G). (2.15)

Proof. The proof follows directly from Theorem 2.16 and Theorem 2.4.

Proposition 2.19. If G is a group of polynomial growth, then there exists 1 [less than or equal to] p < [infinity] such that for any 1-cocycle b: G [right arrow] H we have that

1/[absolute value of g] <b(g), [xi]) [member of] [H.sup.p](G) (2.16)

for all [xi] [member of] H

Proof Fixing a finite generating set S, we have that [[summation].sub.s[member of]S] [parallel]1/[absolute value of g] b(g) - 1/[absolute value of gs] b(gs)[parallel] [much less than] 1/[absolute value of g] choosing an integer p such that [R.sup.p-2] [much greater than] [absolute value of B(R)], we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.17)

from which the result easily obtains.

Conjecture 2.20. If G is a one-ended group of polynomial growth, then for any 1 [less than or equal to] p < [infinity] any positive function f [member of] [H.sup.p](G) belongs to [C.sub.0](G) + C1.

Acknowledgements

We would like to thank Professors Yehuda Shalom and Terence Tao for teaching seminars on the various approaches to Gromov's theorem on groups of polynomial growth in the Fall 2011 quarter at UCLA which stimulated our thoughts in this direction. We are especially grateful to Yehuda Shalom for encouragement. Finally, we thank the anonymous referee for many useful remarks and for pointing us to various relevant results in the literature.

References

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[3] Y. de Cornulier, R. Tessera, and A. Valette, Isometric group actions on Hilbert space: growth of cocycles, Geom. Funct. Anal. 17 (2007), 770-792.

[4] D. Creutz, Ph.D. Thesis, UCLA, June 2011.

[5] P. de la Harpe, Topics in Geometric Group Theory, Chicago Lectures in Mathematics, U. Chicago Press, Chicago, IL, 2000, vi + 310 pp.

[6] P. Delorme, 1-cohomologie et des representations unitaires des groupes de Lie semisimples et resolubles. Produits tensoriels continus de representations, Bull. Soc. Math. France 105 (1977), 281-336.

[7] L. van den Dries and A. J. Wilkie, Gromov's theorem on groups of polynomial growth and elementary logic, J. Algebra 89 (1984), 349-374.

[8] A. Erschler, On isoperimetric profiles of finitely generated groups, Geom. Dedicata 100 (2003), 157-171.

[9] A. Gournay, Vanishing of [l.sup.p]-cohomology and transportation cost, Bull. London Math. Soc. 46 (2014), 481-490.

[10] M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. I. H.E.S. 53 (1981), 53-78.

[11] W. Hebisch and L. Saloff-Coste, Gaussian estimates for Markov chains and random walks on groups, Ann. Prob. 21 (1993), 673-709.

[12] E. Hrushovski, Stable group theory and approximate subgroups, J. Amer. Math. Soc. 25 (2012), 189-243.

[13] B. Kleiner, A new proof of Gromov's theorem on groups of polynomial growth. J. Amer. Math. Soc. 23 (2010), 815-829.

[14] Y. Shalom, Rigidity of commensurators and irreducible lattices, Invent. Math. 141 (2000), 1-54.

[15] Y. Shalom, Harmonic analysis, cohomology, and the large-scale geometry of amenable groups, Acta Math. 192 (2004), 119-185.

[16] Y. Shalom and T. Tao, Afinitary version of Gromov's polynomial growth theorem, Geom. Funct. Anal. 20 (2010), 1502-1547.

[17] R. Tessera: Vanishing of the first reduced cohomology with values in an [L.sup.p]-representation, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 2, 851-876.

[18] R. Tessera, Asymptotic isoperimetry on and uniform embeddings into Banach spaces, Comment. Math. Helv. 86 (2011), 499-535.

[19] R. Tessera, Isoperimetric profile and random walks on locally compact solvable groups, Rev. Mat. Iberoam. 29 (2013), 715-737.

Department of Mathematics, University of Iowa, 14 MacLean Hall, IA 52242, USA and IMAR, Bucharest, Romania

email:ionut-chifan@uiowa.edu

Department of Mathematics, University of California, Los Angeles, Box 951555, Los Angeles, CA 90095-1555, USA

email:thomas.sinclair@math.ucla.edu

Ionut Chifan * Thomas Sinclair ([dagger])

* I.C. was supported in part by NSF Grants #1263982 and #1301370

([dagger]) T.S. was supported by an RTG Assistant Adjunct Professorship

Received by the editors in July 2014.

Communicated by A. Valette.

2010 Mathematics Subject Classification: 22D40; 20F65; 43A15.

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Author: | Chifan, Ionut; Sinclair, Thomas |
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Publication: | Bulletin of the Belgian Mathematical Society - Simon Stevin |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Jul 1, 2015 |

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