Representation growth of quasi-semisimple profinite groups

TL;DR

This paper studies the representation growth of quasi-semisimple profinite groups, characterizing when they have polynomial growth and how their growth degree relates to their structure.

Contribution

It provides a characterization of polynomial representation growth in quasi-semisimple profinite groups and introduces a method to construct groups with prescribed growth degrees.

Findings

Groups with polynomial representation growth are characterized within the class of quasi-semisimple profinite groups.

The growth degree depends only on the semisimple part of the group.

A technique is developed to produce groups with any prescribed positive growth degree.

Abstract

The representation zeta function of a profinite group $G$ encodes the distribution of continuous irreducible complex representations of $G$ as a function of the dimension. Its abscissa of convergence $\alpha(G)$ describes the polynomial degree of representation growth of $G$. Within the class of quasi-semisimple profinite groups, we characterise those of polynomial representation growth (PRG) and we prove that whether such a group $G$ has PRG or not only depends on its semisimple part $G/\mathrm{Z}(G)$. Moreover, we show that, for quasi-semisimple profinite groups $G$ that have uniformly bounded Lie ranks, the degree of growth satisfies $\alpha(G) = \alpha(G/\mathrm{Z}(G))$. We provide a technique to produce, for any prescribed positive real number $\varrho$, quasi-semisimple profinite groups $G$ with PRG of degree $\alpha(G) = \varrho$. Our method allows for considerable flexibility regarding the inclusion of finite simple groups of Lie type as composition factors of $G$. Furthermore, we can arrange for the groups $G$ of prescribed representation growth to be profinite completions of suitable finitely generated discrete groups $\Gamma$ so that the group $\Gamma$ has the same representation zeta function as $G$.