Representation Growth

TL;DR

This thesis investigates the representation growth of various classes of groups, establishing new bounds and counterexamples, and exploring the influence of properties like the Congruence Subgroup Property on growth rates.

Contribution

It extends existing results on polynomial representation growth to all semisimple algebraic groups, provides new bounds on subgroup growth, and constructs counterexamples to the Fake Degree Conjecture.

Findings

Polynomial bounds for representation growth in broader classes of groups

Subgroup growth bounded by n^{D log n} under certain conditions

Counterexamples to the Fake Degree Conjecture involving augmentation ideals

Abstract

The main results in this thesis deal with the representation growth of certain classes of groups. In chapter $1$ we present the required preliminary theory. In chapter $2$ we introduce the Congruence Subgroup Problem for an algebraic group $G$ defined over a global field $k$. In chapter $3$ we consider $\Gamma=G(\mathcal{O}_S)$ an arithmetic subgroup of a semisimple algebraic $k$-group for some global field $k$ with ring of $S$-integers $\mathcal{O}_S$. If the Lie algebra of $G$ is perfect, Lubotzky and Martin showed that if $\Gamma$ has the weak Congruence Subgroup Property then $\Gamma$ has Polynomial Representation Growth, that is, $r_n(\Gamma)\leq p(n)$ for some polynomial $p$. By using a different approach, we show that the same holds for any semisimple algebraic group $G$ including those with a non-perfect Lie algebra. In chapter $4$ we show that if $\Gamma$ has the weak Congruence Subgroup Property then $s_n(\Gamma)\leq n^{D\log n}$ for some constant $D$, where $s_n(\Gamma)$ denotes the number of subgroups of $\Gamma$ of index at most $n$. In chapter $5$ we consider $\Gamma=1+J$, where $J$ is a finite nilpotent associative algebra, this is called an algebra group. We provide counterexamples for any prime $p$ for the Fake Degree Conjecture by looking at groups of the form $\Gamma=1+I_{\mathbb{F}_q}$, where $I_{\mathbb{F}_q}$ is the augmentation ideal of the group algebra $\mathbb{F}_q[\pi]$ for some $p$-group $\pi$. Moreover, we show that for such groups $r_1(\Gamma)=q^{K(\pi)-1}|B_0(\pi)|$, where $B_0(\pi)$ is the Bogomolov multiplier of $\pi$. Finally in chapter $6$, we consider $\Gamma=\prod_{i\in I} S_i$, where the $S_i$ are nonabelian finite simple group. We show that within this class one can obtain any rate of representation growth, i.e., for any $\alpha>0$ there exists $\Gamma=\prod_{i\in I}S_i$ such that $r_n(\Gamma)\sim n^\alpha$.