Sublinear growth of 1-cocycles and uniform convexity

TL;DR

This paper proves that 1-cocycles for certain group representations grow sublinearly, extending results to Banach spaces and deriving implications for strongly mixing unitary representations on Hilbert spaces.

Contribution

It establishes sublinear growth of 1-cocycles for uniformly bounded c0-representations on superreflexive Banach spaces, generalizing known Hilbert space results.

Findings

1-cocycles exhibit sublinear growth in this setting

Extension of Hilbert space results to Banach spaces

Implications for strongly mixing unitary representations

Abstract

Let G be a finitely generated group, let $\pi \colon G \to {\rm GL}(E)$ be a uniformly bounded $c_0$-representation on a superreflexive Banach space $E$, and let $b \colon G \to E$ be a $1$-cocycle for $\pi$. Then $b$ has sublinear growth with respect to the word length. As a corollary we obtain the corresponding Hilbert space statement for strongly mixing unitary representations.