Norms of spherical averaging operators for some geometric group actions

TL;DR

This paper provides asymptotic estimates for the operator norms of spherical averaging operators linked to geometric group actions, especially for Gromov hyperbolic groups, and derives bounds for sphere expansion.

Contribution

It introduces new asymptotic estimates for spherical averaging operators in the context of geometric group actions, with sharp results for hyperbolic groups.

Findings

Asymptotic estimates for $\, ext{ell}^p$-operator norms.

Sharp estimates for Gromov hyperbolic groups.

Lower bounds for combinatorial sphere expansion.

Abstract

We obtain asymptotic estimates for the $\ell^p$-operator norm of spherical averaging operators associated to certain geometric group actions. The motivating example is the case of Gromov hyperbolic groups, for which we obtain asymptotically sharp estimates. We deduce asymptotic lower bounds for the combinatorial expansion of spheres.