Uniformly affine actions on Banach spaces: growth of cocycles

TL;DR

This paper studies the growth of cocycles in uniformly bounded representations on super-reflexive Banach spaces, including $L^p$-spaces and Hilbert spaces, and explores affine actions with optimal growth for certain Property (T) groups.

Contribution

It introduces new results on the growth properties of cocycles in Banach space representations and constructs affine actions with optimal growth for specific groups.

Findings

Growth properties of cocycles in super-reflexive Banach spaces analyzed.

Existence of uniformly Lipschitz affine actions with optimal growth established for $ ext{Sp}(n,1)$ groups.

Generalized Hilbert compression of cocycles studied for Property (T) groups.

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.