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Universidad Autónoma de Madrid Tesis Doctoral Representation Growth Autor: Director: Javier García Rodríguez Andrei Jaikin Zapirain Octubre 2016 a Laura. 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 = D log n G(OS) to show that if Γ has the weak Congruence Subgroup Property then sn(Γ) ≤ 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 k(π)−1 Fq[π] for some p-group π. Moreover, we show that for such groups r1(Γ) = q | B0(π)|, where B0(π) is the Bogomolov multiplier of π. Q Finally in chapter 6, we consider Γ = 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 Q representation growth, i.e., for any α > 0 there exists Γ = i∈I Si where the Si are finite α simple groups of Lie type such that rn(Γ) ∼ n . Moreover, we may take all Si with a fixed Lie type. This complements results of Kassabov and Nikolov in [42]. i Agradecimientos Escribir 100 páginas en 4 años puede no parecer gran cosa. Confieso además que no lo habría podido hacer sin la ayuda de los que me han acompañado estos últimos años. Me gustaría darles las gracias por todas las líneas con las que, de una manera u otra, han contribuido. Gracias en primer lugar a Andrei por estar siempre dispuesto y de buen humor para trabajar, por la barbaridad de cosas que he aprendido y por el placer y la motivación que da trabajar con alguien con tanto talento. Gracias también a Joan por ejercer de hermano mayor. Por los ratos hablando de y pensando en grupos y sus cosas y por todos los buenos ratos que hemos pasado fuera de los grupos y sus cosas. Gracias a Urban porque los meses de trabajo y estudio juntos han sido los más satisfactorios. Gracias a Benjamin por acogerme en Düsseldorf y a toda la gente de allí que me hizo sentir como uno más durante mi estancia. La mejor parte de hacer el doctorado en la UAM ha sido sin duda el grupo de gente con el que he compartido estos años y que han conseguido que madrugara con ganas de irme a la otra punta de Madrid. Empezando por el principio, gracias a Felipe y a Juan por acogerme desde el primer día y ser los mejores tutores de UAM que uno podría tener. Gracias especiales a Ángel, compañero de aventuras desde el primer día y compañero de todos los despachos por los que he pasado. Gracias por tu buen humor, por las charlas matemáticas y no matemáticas y porque siempre apetece echar un café contigo. Gracias en general a toda la cantera de la UAM por aceptar a uno de la Complu sin rechistar (aunque siempre recordándole que la UAM es mejor y más difícil). Desde los veteranos Ana Z, Félix, David T., Bego, Sera, Jose, María, Adri, que ya se fueron a ligas mayores, los que han durado desde el principo hasta el final Carlos, David, Bea P., Felipe, Juan, Irina y los más prometedores Marcos, Bea B, Raquel, Dani, Marta... y también a los mejores fichajes Luis Daniel, Iason, Leyter y Ale. Desayunar, comer (mejor con café y discusión), merendar, cenar, beber, organizar regalos de tesis, celebrar tesis y lo que se terciara ha sido un placer. Mención especial merecen las tardes de fútbol y los Pi Warriors, siempre quedarán en mi memoria. Gracias por aguantarme a mis otros compañeros de despacho Carlos P., Jose Luis, Adri, Jaime y en especial a David por tener el récord de horas de despacho conmigo y servirme de avanzadilla en todo momento, te mereces una Champions (incluso ganándole la final al Madrid). Gracias a mis algebristas preferidos Carlos y Bea por ayudarme con mis eternas dudas de geometría algebraica. Perdón Carlos por las tardes que te he hecho perder y gracias por iii enseñarme con paciencia ese lenguaje tan raro que habláis. Gracias a mis analistas preferidos Jose y Adri por la no numerable cantidad de cosas que he aprendido de vosotros y sobre todo por convencerme de volver a la lucha armada. Gracias a mi matemática preferida, María Medina. Por empezar todas las mañanas con una gran sonrisa y hacer que un mal café sea el mejor de todos. Por reverdecer acelgas por muy pochas que estén, por todas las historias contadas, escritas y por escribir. Gracias a mi Jota, por estar desde siempre y para siempre. Por saber que tengo una amiga que no puedo perder por muchas tesis de 1000 páginas que tenga que hacer. Gracias a mis otros amigos matemáticos, Bruno y Poli, por todos los buenos momentos desde que empezamos a hacer cuentas. Gracias a Fer, el matemático que más me ha influenciado y acompañado desde que empecé, compañero de recomendaciones, amarguras y eternas discusiones, gracias por confiar siempre en mí, especialmente en los malos momentos. Gracias a toda mi familia por apoyarme en todas las etapas hasta llegar hasta aquí. En especial gracias a mis padres y a mi hermana por apoyarme siempre en todo lo que he querido hacer y ayudarme en todo lo que han podido. Gracias con cariño a mis abuelos, por estar siempre orgullosos incluso aunque no entendieran lo que hacía. Y gracias a ti Laura. Por acompañarme y ayudarme en todo desde siempre. Porque sin ti no habría podido, ni esto ni otras muchas cosas. Porque estás 100 páginas también son mérito, esfuerzo y sacrificio tuyo. Por ser la mejor parte de todo desde que tengo memoria. iv Contents Abstract i Agradecimientos iii Introducción 1 Introduction 7 1 Preliminaries 13 1.1 Global and local fields . 13 1.2 Topological groups . 15 1.2.1 Topological groups and group completions . 17 1.2.2 Profinite groups . 18 1.2.3 Analytic groups . 20 1.3 Affine algebraic varieties . 22 1.4 Algebraic groups . 26 1.5 Affine group schemes . 30 1.6 Representation Theory . 31 1.7 Finite groups of Lie type . 33 1.8 Commutative algebra . 33 2 The Congruence Subgroup Problem 35 2.1 Introduction . 35 2.2 Arithmetic groups . 36 2.3 The general formulation . 40 2.4 Reductions and necessary conditions . 41 2.5 The characteristic 0 case . 43 2.6 The semisimple case . 44 3 Representation Growth of Arithmetic Groups 47 3.1 Representation Growth . 47 3.2 CSP and Representation Growth of Arithmetic Groups . 55 v 4 Subgroup Growth of Arithmetic Groups 71 4.1 Subgroup Growth . 71 4.2 CSP and Subgroup Growth of Arithmetic Groups . 73 5 Algebra groups and the Fake Degree Conjecture 77 5.1 Introduction . 77 5.2 Algebra groups and Kirillov’s Orbit Method . 81 5.3 The Fake Degree Conjecture: a disproof . 87 5.4 Abelianization of groups of units and the Bogomolov multiplier . 90 5.4.1 The group (1 + IFq )ab .......................... 90 5.4.2 B0 and rationality questions in unipotent groups . 93 6 Finite groups of Lie type 97 6.1 Introduction . 97 6.2 Representation Growth of Finite Groups of Lie Type . 99 6.3 Polynomial Representation Growth in C .................... 100 6.4 Types of representation growth . 102 vi Introducción Sea Γ un grupo. Una representación de Γ es un homomorfismo de grupos ρ :Γ → GL(V ) para algún espacio vectorial complejo finito dimensional V . La Teoría de Representaciones estudia la relación entre propiedades estructurales de Γ y sus representaciones. El Teorema de Burnside fue la primera gran contribución de la Teoría de Representaciones y desde entonces se ha convertido en un área de investigación en sí misma.
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