2016/12/19 by Javier García-Rodríguez, García-Rodríguez, Javier
Mathematics · #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.1612.06178
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(OS) an arithmetic subgroup of a semisimple algebraic k-group for some global field k with ring of S-integers OS. 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, rn(Γ)≤ 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 sn(Γ)≤ nDlog n for some constant D, where sn(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_\mathbbFq, where I_\mathbbFq is the augmentation ideal of the group algebra \mathbbFq[π] for some p-group π. Moreover, we show that for such groups r1(Γ)=qK(π)-1|B0(π)|, where B0(π) is the Bogomolov multiplier of π. Finally in chapter 6, we consider Γ=∏i∈ I Si, where the Si 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=∏i∈ ISi such that rn(Γ)∼ nα.