Random groups, random graphs and eigenvalues of p-Laplacians

TL;DR

This paper proves fixed point properties for random groups in the triangular density model on $L^p$-spaces for densities above 1/3, using new bounds on the p-Laplacian eigenvalues of random graphs.

Contribution

It establishes fixed point properties for random groups on $L^p$-spaces in the triangular density model and introduces new bounds on the p-Laplacian eigenvalues of random graphs.

Findings

Fixed point properties for random groups on $L^p$-spaces for densities > 1/3.

New bounds on the first eigenvalue of the p-Laplacian on random graphs.

Extension of fixed point results to $L^p$-spaces using eigenvalue bounds.

Abstract

We prove that a random group in the triangular density model has, for density larger than 1/3, fixed point properties for actions on $L^p$-spaces (affine isometric, and more generally $(2-2\epsilon)^{1/2p}$-uniformly Lipschitz) with $p$ varying in an interval increasing with the set of generators. In the same model, we establish a double inequality between the maximal $p$ for which $L^p$-fixed point properties hold and the conformal dimension of the boundary. In the Gromov density model, we prove that for every $p_0 \in [2, \infty)$ for a sufficiently large number of generators and for any density larger than 1/3, a random group satisfies the fixed point property for affine actions on $L^p$-spaces that are $(2-2\epsilon)^{1/2p}$-uniformly Lipschitz, and this for every $p\in [2,p_0]$. To accomplish these goals we find new bounds on the first eigenvalue of the p-Laplacian on random graphs, using methods adapted from Kahn and Szemeredi's approach to the 2-Laplacian. These in turn lead to fixed point properties using arguments of Bourdon and Gromov, which extend to $L^p$-spaces previous results for Kazhdan's Property (T) established by Zuk and Ballmann-Swiatkowski.