Commensurated subgroups of arithmetic groups, totally disconnected groups and adelic rigidity

TL;DR

This paper explores the structure and properties of commensurated subgroups within S-arithmetic groups, connecting to longstanding conjectures in the field and examining their implications for totally disconnected groups and adelic rigidity.

Contribution

It provides new insights and partial resolutions to the Margulis-Zimmer conjecture on commensurated subgroups of S-arithmetic groups.

Findings

Characterization of commensurated subgroups in specific arithmetic contexts

Connections established between subgroup properties and adelic rigidity

Progress towards resolving the Margulis-Zimmer conjecture

Abstract

Investigations into and around a 30-year old conjecture of Gregory Margulis and Robert Zimmer on the commensurated subgroups of S-arithmetic groups.