Representation Growth

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 $r_n(\Gamma)\sim n^\alpha$.