Representation Growth
Author
García Rodríguez, JavierAdvisor
Jaikin Zapirain, AndrésEntity
UAM. Departamento de MatemáticasDate
2016-12-02Subjects
Representaciones de grupos - Tesis doctorales; MatemáticasNote
Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 02-12-2016Esta obra está bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.
Abstract
The main results in this thesis deal with the representation theory of certain classes of groups. More precisely, if rn(Γ) denotes the number of non-isomorphic n-dimensional complex representations of a group Γ, we study the numbers rn(Γ) and the relation of this arithmetic information with structural properties of Γ. 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 in [56] 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 apply our results on representation growth of groups of the form Γ = G(OS) to show that if Γ has the weak Congruence Subgroup Property then sn(Γ) ≤ n D log n for some constant D, where sn(Γ) denotes the number of subgroups of Γ of index at most n. As before, this extends similar results of Lubotzky [54], Nikolov, Abert, Szegedy [1] and Golsefidy [24] for almost simple groups with perfect Lie algebra to any simple algebraic k-group G. In chapter 5 we consider Γ = 1 + J, where J is a finite nilpotent associative algebra, this is called an algebra group. The Fake Degree Conjecture says that for algebra groups the numbers rn(Γ) may be obtained by looking at the square root of the sizes of the orbits arising from Kirillov’s Orbit Method. In [36] Jaikin gave a 2-group that served as a counterexample to this conjecture. We provide counterexamples for any prime p by looking at groups of the form Γ = 1 + IFq , where IFq is the augmentation ideal of the group algebra Fq[π] for some p-group π. Moreover, we show that for such groups r1(Γ) = q k(π)−1 | B0(π)|, where B0(π) is the Bogomolov multiplier of π. Finally in chapter 6, we consider Γ = Q i∈I Si , where the Si are nonabelian finite simple group. We give a characterization of the groups of this form having Polynomial Representation Growth. We also show that within this class one can obtain any rate of representation growth, i.e., for any α > 0 there exists Γ = Q i∈I Si where the Si are finite simple groups of Lie type such that rn(Γ) ∼ n α. Moreover, we may take all Si with a fixed Lie type. This complements results of Kassabov and Nikolov in [42]
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