von-Neumann and Birkhoff ergodic theorems for negatively curved groups

TL;DR

This paper establishes maximal inequalities and a pointwise ergodic theorem for Gromov hyperbolic groups, extending ergodic theory to negatively curved group actions with new maximal inequality results and conditions for spherical shell averages.

Contribution

It introduces maximal inequalities for group averages on hyperbolic groups and proves a pointwise ergodic theorem under specific geometric conditions, advancing ergodic theory in negatively curved spaces.

Findings

Maximal inequalities for ball and shell averages in hyperbolic groups.

A pointwise ergodic theorem for spherical shell averages under certain conditions.

Applicable to groups acting on CAT(-1) spaces.

Abstract

We prove maximal inequalities for concentric ball and spherical shell averages on a general Gromov hyperbolic group, in arbitrary probability preserving actions of the group. Under an additional condition, satisfied for example by all groups acting isometrically and properly discontinuously on CAT(-1) spaces, we prove a pointwise ergodic theorem with respect to a sequence of probability measures supported on concentric spherical shells.