Abstract quotients of profinite groups, after Nikolov and Segal

TL;DR

This paper discusses key results by Nikolov and Segal on the structure of abstract quotients of finitely generated profinite groups, highlighting accessible arguments and core ideas in the area.

Contribution

It provides an accessible overview of Nikolov and Segal's 2012 results on quotients of compact groups, emphasizing the case of finitely generated profinite groups.

Findings

Characterization of abstract quotients of finitely generated profinite groups

Simplified explanations of complex proofs

Insights into the structure of compact Hausdorff groups

Abstract

In this expanded account of a talk given at the Oberwolfach Arbeitsgemeinschaft "Totally Disconnected Groups", October 2014, we discuss results of Nikolay Nikolov and Dan Segal on abstract quotients of compact Hausdorff topological groups, paying special attention to the class of finitely generated profinite groups. Our primary source is a paper, entitled "Generators and commutators in finite groups; abstract quotients of compact groups", that was published by Nikolov and Segal in 2012. Sidestepping all difficult and technical proofs, we present a selection of accessible arguments to illuminate key ideas in the subject.