Compressible subgroups and simplicity

TL;DR

This paper establishes conditions under which groups have simple derived subgroups and normal subgroups are open, extending previous work on micro-supported actions and contributing to the understanding of robustly monolithic groups.

Contribution

It generalizes properties of micro-supported actions to provide new criteria for simplicity and openness of subgroups in locally compact groups.

Findings

Groups with micro-supported actions can have simple derived subgroups.

Non-trivial normal subgroups in certain groups are necessarily open.

Many robustly monolithic groups are shown to be simple-by-discrete.

Abstract

In this article we give sufficient conditions for a group to have simple derived subgroup; the argument is based on generalising properties observed for extremely proximal micro-supported actions on the Cantor space, and generalises previous results of Matui, Le Boudec and others in this direction. We give a sufficient condition for a non-trivial normal subgroup (not assumed closed) of a locally compact group $G$ to be open, also based on the theory of micro-supported actions. This shows in particular that many of the class of robustly monolithic groups introduced by Caprace--Reid--Wesolek are simple-by-discrete.