On common fundamental domains

TL;DR

This paper establishes conditions for two measure-preserving group actions to share a common fundamental domain, with applications to lattices in polynomial growth groups and commuting actions, advancing understanding of fundamental domains in group actions.

Contribution

It provides new criteria for the existence of common fundamental domains for various classes of group actions, including lattices and commuting actions.

Findings

Lattices of equal co-volume in polynomial growth groups have a common fundamental domain.

Conditions are identified for commuting actions to share a fundamental domain.

Results extend to certain semidirect product group actions.

Abstract

We find conditions under which two measure preserving actions of two groups on the same space have a common fundamental domain. Our results apply to commuting actions with separate fundamental domains, lattices in groups of polynomial growth, and some semidirect products. We prove that two lattices of equal co-volume in a group of polynomial growth, one acting on the left, the other on the right, have a common fundamental domain.