## Notes of Romain Tessera’s Cambridge lecture 02-03-2017

A Banachic generalization of Shalom’s property ${H_{FD}}$

With Yves de Cornulier.

1. Shalom’s property ${H_{FD}}$

1.1. Definition

Definition 1 (Shalom)

1. Say a countable discrete group has property ${H_T}$ if a unitary representation has nonzero reduced 1-cohomology, then it contains invariant vectors.
2. Say a countable discrete group has property ${H_T}$ if a unitary representation has nonzero reduced 1-cohomology, then it contains a finite dimensional invariant subspace.

A 1-cocycle defines an affine isometric action. The cocycle is a coboundary iff the corresponding action has a fixed point, or, equivalently, bounded orbits. The cocycle is a limit of coboundaries iff there are almost fixed points, i.e. a sequence of points ${v_n}$ such that for every group element ${g}$, ${d(v_n,gv_n)}$ tends to 0. Hence ${H_T}$ can be reformulated as follows:

In any affine isometric action on a Hilbert space which does not have almost fixed points, the linear part has an invariant vector.

Shalom’s motivation was to give a simpler proof of Gromov’s polynomial growth theorem. He observed that if ${G}$ is amenable and has ${H_{FD}}$, then ${G}$ virtually surjects onto ${{\mathbb Z}}$. Second, he observed that among amenable groups, ${H_{FD}}$ is a quasi-isometry invariant. Indeed, A quasi-isometry of amenable groups induces a bounded measure coupling, cohomology of unitary representations can be induced from one group to the other. Shalom’s program was completed by Ozawa, who proved that groups of polynomial growth have ${H_{FD}}$.

1.2. Applications

Cornulier-Tessera-Valette used ${H_{FD}}$ to give short proofs of classical results:

1. Non-abelian groups do not quasi-isometrically embed into ${L^2}$ (Pauls).
2. ${3}$-regular trees do not quasi-isometrically embed into ${L^2}$ (Bourgain).

For trees, the trick is to first treat the case of the (amenable) lamplighter group, which embeds quasi-isometrically in a product of two trees (it is an undistorted horosphere in this product). Then convert qi embeddings into equivariant qi embeddings (Gromov’s averaging trick).

1.3. Examples

Wreath product ${{\mathbb Z}\wr{\mathbb Z}}$ does not have ${H_{FD}}$.

Lamplighter group ${{\mathbb Z}/p{\mathbb Z}\wr{\mathbb Z}}$ has ${H_{FD}}$.

Theorem 2 (Delorme 1977) Connected solvable Lie groups have ${H_{FD}}$.

Hence virtually polycyclic groups have ${H_{FD}}$.

Question. Is this class qi closed ? Still open.

Unknown wether ${{\mathbb Z}/2{\mathbb Z}\wr{\mathbb Z}^2}$ has ${H_{FD}}$. Shalom also asked wether solvable subgroups of ${Gl({\mathbb Q})}$ have ${H_{FD}}$. Today, among other things, we shall give an answer to this question.

2. Generalization

2.1. WAP representations

Definition 3 A representation of ${G}$ on a Banach space ${E}$ is WAP (for weakly almost periodic) if every orbit is weakly relatively compact.

Example 1 Every representation on a reflexive Banach space is WAP.

Example 2 If ${G}$ acts probability measure preserving on ${X}$, then action on ${L^1(X)}$ is WAP.

The reason that WAP will help us is

Proposition 4 (Alaoglu) If representation ${\pi}$ on ${E}$ is WAP, then invariant vectors have an invariant complement,

$\displaystyle \begin{array}{rcl} E=E^G \oplus W. \end{array}$

This implies that the mean ergodic theorem applies for ${E}$-valued functions.

Example 3 ${{\mathbb Z}}$ has ${WAP_T}$.

Indeed, the displacements of averages of the cocycle tend to 0 thanks to the mean ergodic theorem.

This generalizes to nilpotent groups.

Corollary 5 Non-virtually abelian nilpotent groups do not qi embed into any uniformly convex Banach space (Cheeger-Kleiner).

2.2. Result

Theorem 6 (Cornulier-Tessera) Finitely generated solvable subgroups of ${Gl(d,{\mathbb Q})}$ have ${WAP_{FD}}$.

In fact, our result has a wider scope, including lamplighter group.

This provides again a short proof that 3-regular trees do not qi embed in uniformly convex Banach spaces (which is in fact a characterization of Banach spaces which can be renormed to be uniformly convex).

2.3. Proof

Note that proofs of ${H_T}$ and ${H_{FD}}$ use nondiscrete groups (except Ozawa’s). So does ours. We prove ${WAP_T}$ and ${WAP_{FD}}$ for a class of locally compact solvable groups. Then we show that any solvable subgroup of ${Gl({\mathbb Q})}$ is a cocompact lattice in such a group.

I explain the argument with the semi-direct product ${A={\mathbb R}\times{\mathbb Z}}$ where ${{\mathbb Z}}$ contracts. Let ${S=[-1,1]\times\{t,t^{-1}\}}$ be a compact generating set.

First note that ${A}$ has a controlled Folner sequence ${F_n}$,

$\displaystyle \begin{array}{rcl} \sup_{s\in S}\frac{|F_n\Delta sF_n|}{|F_n|}\leq \frac{C}{n} \end{array}$

and ${F_n\subset S^n}$.

Second, if ${G}$ has a controlled Folner sequence, then a cocycle is a limit of coboundaries iff it is sublinear (in the affine language, orbits are distorted).

Third, every element of ${A}$ can be written ${g=t^{-n}xt^m}$ where ${x\in S}$, ${|n|+|m|\leq |g|_S}$.

Combining the three steps, we see that a cocycle is a limit of coboundaries iff its restriction to the ${{\mathbb Z}}$ factor is.

Fourth, a representation has ${A}$-invariant vectors iff it has ${{\mathbb Z}}$-invariant vectors. Hence ${WAP_T}$ for ${A}$ follows from ${WAP_T}$ for ${{\mathbb Z}}$.

Here is the class of groups for which we prove ${WAP_T}$.

Definition 7 Let ${\mathcal{C}}$ be the class of locally compact groups ${G=UN}$ where

1. ${U}$ is closed and normal,
2. ${N}$ is compact by nilpotent,
3. ${U}$ is an algebraic unipotent subgroups of ${Gl}$ of a product of local fields.
4. ${U}$ is generated by finitely many subgroups, each of which is contracted by some element of ${N}$

We prove ${WAP_{FD}}$ for a larger class of groups made from cocompact subgroups of the previous class. This class contains all solvable Lie groups, and a bit more, enough to accomodate solvable groups of rational matrices.