** A Banachic generalization of Shalom’s property **

With Yves de Cornulier.

**1. Shalom’s property **

** 1.1. Definition **

Definition 1 (Shalom)

- Say a countable discrete group has property if a unitary representation has nonzero reduced 1-cohomology, then it contains invariant vectors.
- Say a countable discrete group has property 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 such that for every group element , tends to 0. Hence 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 is amenable and has , then virtually surjects onto . Second, he observed that among amenable groups, 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 .

** 1.2. Applications **

Cornulier-Tessera-Valette used to give short proofs of classical results:

- Non-abelian groups do not quasi-isometrically embed into (Pauls).
- -regular trees do not quasi-isometrically embed into (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 does not have .

Lamplighter group has .

Theorem 2 (Delorme 1977)Connected solvable Lie groups have .

Hence virtually polycyclic groups have .

**Question**. Is this class qi closed ? Still open.

Unknown wether has . Shalom also asked wether solvable subgroups of have . Today, among other things, we shall give an answer to this question.

**2. Generalization **

** 2.1. WAP representations **

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

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

Example 2If acts probability measure preserving on , then action on is WAP.

The reason that WAP will help us is

Proposition 4 (Alaoglu)If representation on is WAP, then invariant vectors have an invariant complement,

This implies that the mean ergodic theorem applies for -valued functions.

Example 3has .

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

This generalizes to nilpotent groups.

Corollary 5Non-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 have .

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 and use nondiscrete groups (except Ozawa’s). So does ours. We prove and for a class of locally compact solvable groups. Then we show that any solvable subgroup of is a cocompact lattice in such a group.

I explain the argument with the semi-direct product where contracts. Let be a compact generating set.

First note that has a controlled Folner sequence ,

and .

Second, if 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 can be written where , .

Combining the three steps, we see that a cocycle is a limit of coboundaries iff its restriction to the factor is.

Fourth, a representation has -invariant vectors iff it has -invariant vectors. Hence for follows from for .

Here is the class of groups for which we prove .

Definition 7Let be the class of locally compact groups where

- is closed and normal,
- is compact by nilpotent,
- is an algebraic unipotent subgroups of of a product of local fields.
- is generated by finitely many subgroups, each of which is contracted by some element of

We prove 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.