This thesis is devoted to the study of operators on discrete structures. The operators are supposed to be self-adjoint and obey a certain translation invariance property. The discrete structures are given as Cayley graphs via finitely generated groups. Here, sofic groups and amenable groups are in the center of our considerations. Note that every finitely generated amenable group is sofic. We investigate the spectrum of a discrete self-adjoint operator by studying a sequence of finite dimensional analogues of these operators. In the setting of amenable groups we obtain these approximating operators by restricting the operator in question to finite subsets Qn , n ∈ N. These finite dimensional operators are self-adjoint and therefore admit a well-defined normalized eigenvalue counting function. The limit of the normalized eigenvalue counting functions when |Qn | → ∞ (if it exists) is called the integrated density of states (IDS). It is a distribution function of a probability measure encoding the distribution of the spectrum of the operator in question on the real axis. In this thesis, we prove the existence of the IDS in various geometric settings and for different types of operators. The models we consider include deterministic as well as random situations. Depending on the specific setting, we prove existence of the IDS as a weak limit of distribution functions or even as a uniform limit. Moreover, in certain situations we are able to express the IDS via a semi-explicit formula using the trace of the spectral projection of the original operator. This is sometimes referred to as the validity of the Pastur-Shubin trace formula. In the most general geometric setting we study, the operators are defined on Cayley graphs of sofic groups. Here we prove weak convergence of the eigenvalue counting functions and verify the validity of the Pastur-Shubin trace formula for random and non-random operators . These results apply to operators which not necessarily bounded or of finite hopping range. The methods are based on resolvent techniques. This theory is established without having an ergodic theorem for sofic groups at hand. Note that ergodic theory is the usual tool used in the proof of convergence results of this type. Specifying to operators on amenable groups we are able to prove stronger results. In the discrete case, we show that the IDS exists uniformly for a certain class of finite hopping range operators. This is obtained by using a Banach space-valued ergodic theorem. We show that this applies to eigenvalue counting functions, which implies their convergence with respect to the Banach space norm, in this case the supremum norm. Thus, the heart of this theory is the verification of the Banach space-valued ergodic theorem. Proceeding in two steps we first prove this result for so-called ST-amenable groups. Then, using results from the theory of ε-quasi tilings, we prove a version of the Banach space-valued ergodic theorem which is valid for all amenable groups. Focusing on random operators on amenable groups, we prove uniform existence of the IDS without the assumption that the operator needs to be of finite hopping range or bounded. Moreover, we verify the Pastur-Shubin trace formula. Here we present different techniques. First we show uniform convergence of the normalized eigenvalue counting functions adapting the technique of the Banach space-valued ergodic theorem from the deterministic setting. In a second approach we use weak convergence of the eigenvalue counting functions and additionally obtain control over the convergence at the jumps of the IDS. These ingredients are applied to verify uniform existence of the IDS. In both situations we employ results from the theory of large deviations, in order to deal with long-range interactions.

Ce mémoire porte sur l’étude du laplacien sur des surfaces de Riemann. En particulier, nous nous intéressons à ses valeurs propres qui représentent les notes que jouerait la surface si elle était un tambour. Les valeurs les plus étudiées sont la première valeur propre non nulle λ1 ainsi que sa multiplicité m1 (la dimension de l’espace propre). Notamment, Colin de Verdière conjecturait que m1 est toujours borné supérieurement par le nombre chromatique moins 1. Des travaux de Fortier Bourque et Petri ont montré que parmi toutes les surfaces hyperboliques de genre 3, c’est la quartique de Klein qui maximise la multiplicité et atteint la borne supérieure conjecturée par Colin de Verdière. Cette surface est la première d’une suite de surfaces hautement symétriques, les surfaces de Hurwitz. Nous montrons à l’aide de la formule des traces de Selberg que pour la prochaine surface dans la suite, la surface de Fricke–Macbeath F, nous avons m1(F) = 7. Une recherche indépendante menée par Chul-hee Lee arrive au même résultat à propos de la multiplicité. Le chapitre 1 introduit des notions géométriques comme la géométrie hyperbolique, les surfaces hyperboliques et triangulaires ainsi que le théorème de Hurwitz. Le chapitre 2 présente des concepts de base de théorie spectrale ainsi que des outils comme la formule des traces de Selberg et la théorie de la représentation. Le chapitre 3 est dédié à l’étude de la surface de Fricke–Macbeath et à la preuve de notre résultat principal à l’aide des outils des chapitres précédents. Dans le chapitre 4, nous discutons de nouvelles techniques de calcul de m1 qui ont été utilisées pour montrer l’existence de contre-exemples à la conjecture de Colin de Verdière dans des travaux conjoints avec Fortier Bourque, Gruda-Mediavilla et Petri. / This master’s thesis studies the Laplace operator on Riemann surfaces. We are especially interested in its eigenvalues, which correspond to the notes that the surface would play if it were a drum. In particular, the first non-zero eigenvalue λ1 and its multiplicity m1 (the dimension of the corresponding eigenspace) have been well studied. For instance, Colin de Verdière conjectured that m1 is bounded above by the chromatic number minus 1 based on a few examples. Later work by Fortier Bourque and Petri showed that among hyperbolic surfaces of genus 3, the Klein quartic maximizes the multiplicity, and attains the upper bound conjectured by Colin de Verdière. This surface is the first of a sequence of highly symmetrical surfaces named Hurwitz surfaces. We will show using the Selberg trace formula that for the next surface in the sequence, the Fricke–Macbeath surface F, we have m1(F) = 7. This result was also obtained independently by Chul-hee Lee. Chapter 1 introduces some geometric notions including hyperbolic geometry, hyperbolic surfaces, and triangular surfaces, followed by Hurwitz’s automorphism theorem. Chapter 2 covers some basic concepts in spectral theory as well as some useful tools like the Selberg trace formula and a bit of representation theory. Chapter 3 focuses on the study of the Fricke–Macbeath surface and the proof of our main result using the techniques introduced in previous chapters. Finally, Chapter 4 discusses new methods for computing m1 which were used to show the existence of counterexamples to Colin de Verdière’s conjecture in joint work with Fortier Bourque, Gruda-Mediavilla, and Petri.