About discrete subgroups of full groups of measure preserving equivalence relations

TL;DR

This paper investigates the structure of countable subgroups within full groups of measure-preserving equivalence relations, establishing conditions that prevent these subgroups from being topologically discrete and discussing related conjectures.

Contribution

It introduces new constraints on subgroup structure and action properties that imply non-discreteness of the subgroup topology in full groups.

Findings

Certain subgroup properties imply the subgroup topology is not discrete

Constraints on the measure of fixed point sets influence subgroup topology

Discussion of conjectures regarding discrete subgroups in full groups

Abstract

In this note we study countable subgroups of the full group of a measure preserving equivalence relation. We provide various constraints on the group structure, the nature of the action, and on the measure of fixed point sets, that imply that the subgroup topology is not discrete. We mention various conjectures about discrete subgroups of full groups.