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 $Γ=G(\mathcal{O}_S)$ 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 $Γ$ has the weak Congruence Subgroup Property then $Γ$ has Polynomial Representation Growth, that is, $r_n(Γ)\leq p(n)$ 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 $Γ$ has the weak Congruence Subgroup Property then $s_n(Γ)\leq n^{D\log n}$ for some constant $D$, where $s_n(Γ)$ denotes the number of subgroups of $Γ$ of index at most $n$. In chapter $5$ we consider $Γ=1+J$, 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 $Γ=1+I_{\mathbb{F}_q}$, where $I_{\mathbb{F}_q}$ is the augmentation ideal of the group algebra $\mathbb{F}_q[π]$ for some $p$-group $π$. Moreover, we show that for such groups $r_1(Γ)=q^{K(π)-1}|B_0(π)|$, where $B_0(π)$ is the Bogomolov multiplier of $π$. Finally in chapter $6$, we consider $Γ=\prod_{i\in I} S_i$, 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 $α>0$ there exists $Γ=\prod_{i\in I}S_i$ such that $r_n(Γ)\sim n^α$.