Geometric covering arguments and ergodic theorems for free groups

TL;DR

This paper introduces a geometric covering method to prove ergodic theorems for free group actions, extending classical techniques from amenable groups to a broader class of averages.

Contribution

It develops a new geometric approach for ergodic theorems in free groups, generalizing previous results and enabling analysis of new types of averages.

Findings

Extended maximal and pointwise ergodic theorems for free groups.

Applicable to a large class of geometric averages.

Brief discussion on applications to other groups.

Abstract

We present a new approach to the proof of ergodic theorems for actions of free groups based on geometric covering and asymptotic invariance arguments. Our approach can be viewed as a direct generalization of the classical geometric covering and asymptotic invariance arguments used in the ergodic theory of amenable groups. We use this approach to generalize the existing maximal and pointwise ergodic theorems for free group actions to a large class of geometric averages which were not accessible by previous techniques. Some applications of our approach to other groups and other problems in ergodic theory are also briefly discussed.