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.


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