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- W4293488987 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 $s_n(Gamma)leq n^{Dlog n}$ 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 $Gamma=prod_{iin I} S_i$, 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 $Gamma=prod_{iin I}S_i$ such that $r_n(Gamma)sim n^alpha$." @default.
- W4293488987 created "2022-08-29" @default.
- W4293488987 creator A5089668583 @default.
- W4293488987 date "2016-12-19" @default.
- W4293488987 modified "2023-09-27" @default.
- W4293488987 title "Representation Growth" @default.
- W4293488987 doi "https://doi.org/10.48550/arxiv.1612.06178" @default.
- W4293488987 hasPublicationYear "2016" @default.
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