A Banachic generalization of Shalom’s property
With Yves de Cornulier.
1. Shalom’s property
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 .
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).
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.
2.1. WAP representations
Definition 3 A representation of on a Banach space 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 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 3 has .
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).
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).
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 ,
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 7 Let 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.