Banach Zuk's criterion for partite complexes with application to random groups

TL;DR

This paper extends Zuk's criterion to Banach spaces for groups acting on partite complexes, leading to new fixed point results for random groups in the Gromov density model across various Banach space classes.

Contribution

It introduces a Banach space version of Zuk's criterion and applies it to establish fixed point properties for random groups in the Gromov density model.

Findings

Groups in the Gromov density model have property (F L^p) asymptotically almost surely.

Provides a sharp lower bound for the conformal dimension growth of boundary groups.

Establishes fixed point theorems for groups acting on Banach spaces.

Abstract

We prove a Banach version of \.Zuk's criterion for groups acting on partite simplicial complexes. Using this new criterion we derive a new fixed point theorem for random groups in the Gromov density model with respect to several classes of Banach spaces ($L^p$ spaces, Hilbertian spaces, uniformly curved spaces). In particular, we show that for every $p$, a group in the Gromov density model has asymptotically almost surely property $(F L^p)$ and give a sharp lower bound for the growth of the conformal dimension of the boundary of such group as a function of the parameters of the density model.