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Uniformly affine actions on Banach spaces: growth of cocycles

Kevin Boucher, Georg Grutzner

Abstract

We investigate growth properties of cocycles with values in uniformly bounded representations on super-reflexive Banach spaces; this includes $L^p$-spaces for $1<p<\infty$ as well as Hilbert spaces. We then study the generalized Hilbert compression of cocycles arising in this setting for the Property (T) groups $\mathrm{Sp}(n,1)$, $n\ge 2$, and establish the existence of uniformly Lipschitz affine actions with optimal growth.

Uniformly affine actions on Banach spaces: growth of cocycles

Abstract

We investigate growth properties of cocycles with values in uniformly bounded representations on super-reflexive Banach spaces; this includes -spaces for as well as Hilbert spaces. We then study the generalized Hilbert compression of cocycles arising in this setting for the Property (T) groups , , and establish the existence of uniformly Lipschitz affine actions with optimal growth.
Paper Structure (10 sections, 10 theorems, 60 equations)

This paper contains 10 sections, 10 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Statement of the results
  3. Results
  4. Organisation of the paper
  5. Acknowledgements
  6. Optimal compression exponent
  7. A smooth version of the harmonic representation of cocycles
  8. The Hilbert space case
  9. Real rank 1 - Proof of Theorem \ref{['6ea617']}
  10. Proof of Corollary \ref{['99b6bf']}

Key Result

Theorem 1.4

Let $\Gamma$ be a finitely generated, discrete group and $S$ a finite generating set of $\Gamma$. Let $(\pi,B)$ be uniformly bounded representation on a $p$-uniformly smooth Banach space $B$. If $\Gamma$ admits a cocycle, $b$, with coefficients in $(\pi,B)$ such that then $\Gamma$ is amenable.

Theorems & Definitions (29)

  • Conjecture 1.1: Shalom
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Corollary 1.7
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 19 more