Counting irreducible modules for profinite groups

TL;DR

This paper studies the representation growth of profinite groups over finite fields, characterizing groups with bounded exponential growth and exploring their structural properties and examples.

Contribution

It provides necessary and sufficient conditions for groups to have uniformly bounded exponential representation growth and constructs examples with unexpected properties.

Findings

UBERG groups are closed under split extensions

Profinite groups of type FP_1 with UBERG are finitely generated

Constructed examples of groups with unusual properties

Abstract

This article is concerned with the representation growth of profinite groups over finite fields. We investigate the structure of groups with uniformly bounded exponential representation growth (UBERG). Using crown-based powers we obtain some necessary and some sufficient conditions for groups to have UBERG. As an application we prove that the class of UBERG groups is closed under split extensions but fails to be closed under extensions in general. On the other hand, we show that the closely related probabilistic finiteness property $PFP_1$ is closed under extensions. In addition, we prove that profinite groups of type $FP_1$ with UBERG are always finitely generated and we characterise UBERG in the class of pro-nilpotent groups. Using infinite products of finite groups, we construct several examples of profinite groups with unexpected properties: (1) an UBERG group which cannot be finitely generated, (2) a group of type $PFP_\infty$ which is not UBERG and not finitely generated and (3) a group of type $PFP_\infty$ with superexponential subgroup growth.