Random groups, random graphs and eigenvalues of p-Laplacians

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.

v1: 51 pages; v2: 54 pages, minor updates and corrections to 2<p<3 case; v3: 56 pages, minor updates

References in corpus (2)

• Freiheitssatz and phase transition for the density model of random groups
• A note on quotients of word hyperbolic groups with Property (T)

Cited by in corpus (6)

• Acylindrically hyperbolic groups with exotic properties
• Quasi-isometric diversity of marked groups
• Conformal dimension of hyperbolic groups that split over elementary subgroups
• Garland's method with Banach coefficients
• Random triangular Burnside groups
• On the l^{q,p} cohomology of Carnot groups