Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2001. / Includes bibliographical references (p. 112).

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1999. / Includes bibliographical references. Also available on the Internet.

The classical spherical and Euclidean geometries are easy to visualize and correspond to spaces with constant curvature 0 and +1 respectively. The geometry with constant curvature −1, hyperbolic geometry, is much more complex. A powerful theorem of Mostow and Prasad states that in all dimensions at least 3, the geometry of a finite-volume hyperbolic manifold (a space with local d-dimensional hyperbolic geometry) is determined by the manifold's fundamental group (a topological invariant of the manifold). A cusp is a part of a finite-volume hyperbolic manifold that is infinite but has finite volume (cf. the surface of revolution of a tractrix has finite area but is infinite). All non-compact hyperbolic manifolds have cusps, but only finitely many of them. In the fundamental group of such a manifold, each cusp corresponds to a cusp subgroup, and each cusp subgroup is associated to a point on the boundary of H^d, which can be identified with the (d − 1)-sphere. It is known that there are many one-cusped two- and three-dimensional hyperbolic manifolds. This thesis studies restrictions on the existence of 1-cusped hyperbolic d-dimensional manifolds for d ≥ 3. Congruence subgroups belong to a special class of hyperbolic manifolds called arithmetic manifolds. Much is known about arithmetic hyperbolic 3- manifolds, but less is known about arithmetic hyperbolic manifolds of higher dimensions. An important infinite class of arithmetic d-manifolds is obtained using SO(n, 1; Z), a subset of the integer matrices with determinant 1. This is known to produce 1-cusped examples for small d. Taking special congruence conditions modulo a fixed number, we obtain congruence subgroups of SO(n, 1; Z) which also have cusps but possibly more than one. We ask what congruence subgroups with one cusp exist in SO(n, 1; Z). We consider the prime congruence level case, then generalize to arbitrary levels. Covering space theory implies a relation between the number of cusps and the image of a cusp in the mod p reduced group SO(d+ 1, p), an analogue of the classical rotation Lie group. We use the sizes of maximal subgroups of groups SO(d + 1, p), and the maximal subgroups' geometric actions on finite vector spaces, to bound the number of cusps from below. Let Ω(d, 1; Z) be the index 2 subgroup in SO(d, 1; Z) that consists of all elements of SO(d, 1; Z) with spinor norm +1. We show that for d = 5 and d ≥ 7 and all q not a power of 2, there is no 1-cusped level-q congruence subgroup of Ω(d, 1; Z). For d = 4, 6 and all q not of the form 2^a3^b, there is no 1-cusped level-q congruence subgroup of Ω(d, 1; Z). / Mathematics

n previous work, the author and his collaborators developed a relationship in the SL(2,C) representation theories of two finitely generated groups with isomorphicprofinite completions assuming a certain strong representation rigidity for one of thegroups. This was then exploited as one part of producing examples of lattices in SL(2,C) which are profinitely rigid. In this article, the relationship is extended to representations in any connected reductive algebraic groups under a weaker representation rigidity hypothesis. The results are applied to lattices in higher rank Liegroups where we show that for some such groups, including SL(n,Z) forn≥3, they are either profinitely rigid, or they contain a proper Grothendieck subgroup.

Le sujet principal de ce mémoire est le comportement asymptotique de la géométrie et topologie des variétés localement symétriques Gamma\ X quand le volume tend vers l’infini. Notre premier résultat porte sur la convergence Benjamini-Schramm des 2 ou 3-variétés hyperboliques arithmétiques. Une suite d'espaces localement symétriques (Gamma_n\ X) converge Benjamini-Schramm vers l'espace symétrique X si pour chaque R>0 la limite de \Vol((\Gamma\X)_{<R})/Vol(\Gamma\bs X). On montre qu'il existe une constante réelle C=C_R satisfaisant la propriété suivante: pour chaque réseau arithmétique de congruence Gamma de \PGL(2,R) ou PGL(2,C) sans torsion on a Vol ((Gamma\ X)_{<R})<= C_R \ Vol (Gamma\ X)^0.986. Il n'y a qu'un nombre fini de réseaux arithmétiques de covolume borné par une constante donc ce résultat implique la convergence Benjamini-Schramm pour des variétés arithmétiques de congruence. On donne aussi une version de (\ref{AbsFr1}) un peu plus faible qui reste vraie pour des réseaux arithmétiques qui ne sont pas de congruence. Les majorations de volume de la partie $R$-mince sont déduites d'une version forte de la propriété de la multiplicité limite satisfaite par les réseaux arithmétiques de PGL(2,R) et PGL(2,C). En utilisant nos résultats on confirme la conjecture de Gelander pour des 3-variétés arithmétiques hyperboliques: pour chaque telle variété M on construit un complexe simplicial N homotope à M dont le nombre des simplexes est O(Vol(M)) et le degré des nœuds est uniformément borné par une constante absolue. Dans la deuxième partie on s'intéresse aux espaces localement symétriques Gamma\X où X est de rang supérieur ou égal à 2. Notre résultat principal affirme que la dimension du premier groupe d'homologie à coefficients dans F_2 (corps avec 2 éléments) est sous-linéaire en le volume. Ce résultat est à comparer avec des travaux de Calegari et Emerton sur la cohomologie mod-p dans les tours p-adiques des 3-variétés et les résultats d'Abert, Gelander et Nikolov sur le rang des sous-groupes d'un réseau de rang supérieur à angles droits. Le point fort de notre approche est qu'il n'y a pas besoin de travailler dans une seule classe de commensurabilité. La troisième partie est indépendante des deux premières. Elle porte sur une extension du théorème de Kesten. Le théorème de Kesten affirme que si Gamma est un groupe engendré par un ensemble fini symétrique S, N est un sous-groupe normal de Gamma alors N est moyennable si et seulement si les rayons spectraux du graphe de Cayley Cay(Gamma,S) et du graphe de Scheier Sch(Gamma/N,S) coïncident. En utilisant les techniques de Abert, Glasner et Virag on généralise le theorème de Kesten aux N-uniformément récurrents. / The main theme of this work is the study of geometry and topology of locally symmetric spaces Gamma\ X as ther volume Vol(\Gamma\ X) tends to infinity. Our first main result concerns the Benjamini-Schramm convergence for arithmetic hyperbolic 2 or 3-manifolds. A sequence of locally symmetric spaces (Gamma_n\ X) converges Benjamini-Schramm to X if and only if for every radius R>0 the limit Vol((Gamma\ X)_{<R}/Vol (Gamma\ X) as n goes to infinity is 0, where (\Gamma\X)_{<R} stands for the R-thin part of Gamma\ X. We prove that there exists a positive constant C=C_R with the following property: for every torsion free, uniform, congruence arithmetic lattice Gamma in PGL(2,R) or PGL(2,C) Vol ((Gamma\ X)_{<R})<= C Vol (Gamma\X))^0.986. There is only finitely many arithmetic lattices of covolume bounded by a constant so the result above implies the Benjamini-Schramm convergence for any sequence of congruence arithmetic hyperbolic 3-manifolds. We also prove a similar but slightly weaker inequality for non-congruence subgroups. Our results are deduced form a strong form of the limit multiplicity property that holds for arithmetic lattices in PGL(2,R) of PGL(2,C). As an application of our bounds we confirm Gelander's conjecture on the triangulations of arithmetic hyperbolic 3-manifolds: we show that every arithmetic hyperbolic 3-manifold M admits a triangulation with O(Vol(M)) simplices and degrees of vertices bounded uniformly by an absolute constant. Next, we move to the setting of higher rank locally symmetric spaces. Let M_n=Gamma_n\ X be a sequence of pairwise distinct locally symmetric spaces modeled after a higher rank symmetric space X. We show that the dimension of the first homology group with coefficients in F_2 is sublinear in volume. This can be compared with the results of Calegari and Emerton on mod-p homology growth in p-adic analytic towers of 3-manifolds as well as the results of Abert, Gelander and Nikolov on the rank gradient of right-angled lattices in higher rank Lie groups.The main strength of our theorem is that we do not need to assume that the manifolds in question are commensurable. Our third result is independent of the first two. Kesten theorem asserts that if Gamma is group generated by a finite symmetric set S and N is a normal subgroup of Gamma then N is amenable if and only if the spectral radii of the Cayley graphs Cay(Gamma, S) and the Schreier graph Sch(Gamma/N,S) are equal. Building on the work of Abert, Glasner and Virag we extend Kesten's theorem to uniformly recurrent subgroups.