Pointwise ergodic theorems beyond amenable groups

TL;DR

This paper extends pointwise and maximal ergodic theorems to a broad class of non-amenable groups by leveraging amenable actions and equivalence relations, covering all irreducible lattices in certain Lie groups.

Contribution

It introduces a novel approach to ergodic theorems for non-amenable groups via amenable actions and equivalence relations, expanding the scope beyond traditional amenable groups.

Findings

Proved ergodic theorems for groups with specific amenable actions.

Extended ergodic theory to include amenable equivalence relations.

Applicable to all irreducible lattices in connected semisimple Lie groups.

Abstract

We prove pointwise and maximal ergodic theorems for probability measure preserving (p.m.p.) actions of any countable group, provided it admits an essentially free, weakly mixing amenable action of stable type $III_1$. We show that this class contains all irreducible lattices in connected semisimple Lie groups without compact factors. We also establish similar results when the stable type is $III_\lambda$, $0 < \lambda < 1$, under a suitable hypothesis. Our approach is based on the following two principles. First, we show that it is possible to generalize the ergodic theory of p.m.p. actions of amenable groups to include p.m.p. amenable equivalence relations. Second, we show that it is possible to reduce the proof of ergodic theorems for p.m.p. actions of a general group to the proof of ergodic theorems in an associated p.m.p. amenable equivalence relation, provided the group admits an amenable action with the properties stated above.