Strange images of profinite groups

TL;DR

This paper explores the properties of finitely generated profinite groups, demonstrating that certain types cannot have finitely generated infinite images and examining the existence of dense normal subgroups.

Contribution

It extends understanding of the structure of profinite groups by proving the non-existence of finitely generated infinite images in specific classes and analyzing dense normal subgroups.

Findings

Finitely generated profinite groups cannot have finitely generated infinite images if they are semisimple or nonuniversal.

The paper confirms Segal's result for prosoluble groups and extends it to other classes.

Dense normal subgroups are investigated within these groups.

Abstract

We investigate whether a finitely generated profinite group G could have a finitely generated infinite image. A result of Dan Segal shows that this is impossible if G is prosoluble. We prove that such an image does not exist if G is semisimple or nonuniversal. We also investigate the existense of dense normal subgroups in $G$.