On growth of cocycles of isometric representations on $L^p$-spaces

TL;DR

This paper investigates the growth behavior of 1-cocycles in isometric representations on Banach spaces, revealing a dichotomy between boundedness and rapid growth, with implications for fixed point properties and Liouville properties.

Contribution

It establishes a dichotomy for 1-cocycles on groups acting on $L^p$-spaces, showing either boundedness or existence of fast-growing cocycles, extending previous work inspired by Lafforgue.

Findings

All 1-cocycles are bounded or exhibit fast growth in $L^p$-spaces.

Bounds on average growth of harmonic 1-cocycles in Banach spaces.

Existence of cocycles with large compression implies Liouville property.

Abstract

We study different notions of asymptotic growth for 1-cocycles of isometric representations on Banach spaces. One can see this as a way of quantifying the absence of fixed point properties on such spaces. Inspired by the work of Lafforgue, we show the following dichotomy: for a compactly generated group $G$, either all 1-cocycles of $G$ taking values in $L^p$-spaces are bounded (this is Property $FL^p$) or there exists a 1-cocycle of $G$ taking values in an $L^p$-space with relatively fast growth. We also obtain upper and lower bounds on the average growth of harmonic 1-cocycles with values in Banach spaces with convexity properties. As a consequence, we obtain bounds on the average growth of all 1-cocycles with values in $L^p$-spaces for groups with property $(T)$. Lastly, we show that for a compactly generated group $G$, the existence of a 1-cocycle with compression larger than $\sqrt{n}$ implies the Liouville property for a large family of probability measures on $G$.