Infinite groups from the profinite point of view

TL;DR

This paper surveys recent research on how the structure and properties of infinite groups can be understood through their finite quotients, with a focus on $S$-arithmetic and branch groups.

Contribution

It provides a comprehensive overview of current developments in understanding infinite groups via their finite quotients, highlighting key classes like $S$-arithmetic and branch groups.

Findings

Finite quotients often determine group properties

$S$-arithmetic groups exhibit specific quotient behaviors

Branch groups have unique finite quotient structures

Abstract

We survey recent work ranging around the question in how far a group, or a property of a group, is determined by the set of finite quotient groups. Our focus lies on $S$-arithmetic groups, branch groups, and their relatives.