This dissertation is devoted to the study of self-similar groups and related topics. It consists of three parts. The first part is devoted to the study of examples of finitely generated amenable groups for which every finitely presented cover contains non-abelian free subgroups. The study of these examples was motivated by natural questions about finiteness properties of finitely generated groups. We show that many examples of amenable self-similar groups studied in the literature cannot be covered by finitely presented amenable groups. We investigate the class of contracting self-similar groups from this perspective and formulate a general result which is used to detect this property. As an application we show that almost all known examples of groups of intermediate growth cannot be covered by finitely presented amenable groups. The latter is related to the problem of the existence of finitely presented groups of intermediate growth. The second part focuses on the study of one important example of a self-similar group called the first Grigorchuk group G, from the viewpoint of pro finite groups. We investigate finite quotients of this group related to presentations and group (co)homology. As an outcome of this investigation we prove that the pro finite completion G_hat of this group is not finitely presented as a pro finite group. The last part focuses on a class of recursive group presentations known as L-presentations, which appear in the study of self-similar groups. We investigate the relation of such presentations with the normal subgroup structure of finitely presented groups and show that normal subgroups with finite cyclic quotient of finitely presented groups have such presentations. We apply this result to finitely presented indicable groups without free sub-semigroups.

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Let G be a locally compact group, w be a continuous weight function on G, and L^1(G,w) be the corresponding Beurling algebra. In this thesis, we study weak amenability of L^1(G,w) and of its centre ZL^1(G,w) for non-commutative locally compact groups G. We first give examples to show that the condition that characterizes weak amenability of L^1(G,w) for commutative groups G is no longer sufficient for the non-commutative case. However, we prove that this condition remains necessary for all [IN] groups G. We also provide a necessary condition for weak amenability of L^1(G,w) of a different nature, which, among other things, allows us to obtain a number of significant results on weak amenability of l^1(F_2,w) and l^1((ax+b),w). We then study the relation between weak amenability of the algebra L^1(G,w) on a locally compact group G and the algebra L^1(G/H,^w) on the quotient group G/H of G over a closed normal subgroup H with an appropriate weight ^w induced from w. We give an example showing that L^1(G,w) may not be weakly amenable even if both L^1(G/H,^w) and L^1(H,w|_H) are weakly amenable. On the other hand, by means of constructing a generalized Bruhat function on G, we establish a sufficient condition under which weak amenability of L^1(G,w) implies that of L^1(G/H,^w). In particular, with this approach, we prove that weak amenability of the tensor product of L^1(G_1,w_1) and L^1(G_2,w_2) implies weak amenability of both Beurling algebras L^1(G_1,w_1) and L^1(G_2,w_2), provided the weights w_1, w_2 are bounded away from zero. However, given a general weight on the direct product G of G_1 and G_2, weak amenability of L^1(G,w) usually does not imply that of L^1(G_1,w|_{G_1}), even if both G_1, G_2 are commutative. We provide an example to illustrate this. While studying the centres ZL^1(G,w) of L^1(G,w), we characterize weak amenability of ZL^1(G,w) for connected [SIN] groups G, establish a necessary condition for weak amenability of ZL^1(G,w) in the case when G is an [FC] group, and give a sufficient condition for the case when G is an [FD] group. In particular, we obtain some positive results on weak amenability of ZL^1(G,w) for a compactly generated [FC] group G with a polynomial weight w. Finally, we briefly discuss the derivation problem for weighted group algebras and present a partial solution to it.

Cette thèse étudie la moyennabilité et la propriété de Liouville des groupes pleins-topologiques des systèmes de Cantor, des groupes d'échanges d'intervalles, et des groupes agissants sur les arbres enracinés. Dans le Chapitre 2, nous obtenons les premiers exemples de groupes simples, infinis, de type fini, tels que le bord de Poisson de toute marche aléatoire simple est trivial (la propriété de Liouville). Ces exemples sont des sous-groupes dérivés de groupes pleins topologiques d'une famille de sous-décalages minimaux. Nous montrons que si la complexité d'un sous-décalage (pas nécessairement minimal) est strictement sous-quadratique, toute mesure de probabilité symétrique de support fini sur le groupe plein-topologique est d'entropie asymptotique nulle. Dans le Chapitre 3, nous exhibons une famille de groupes pleins-topologiques de sous-décalages minimaux qui contiennent les groupes de Grigorchuk G_ω comme sous-groupes. Cette construction montre que le groupe plein-topologique d'un sous-décalage minimal peut avoir des sous-groupes de croissance intermédiaire, en répondant à une question de Grigorchuk. Dans le Chapitre 4 (basé sur un travail en commun avec K. Juschenko, N. Monod, M. de la Salle) nous étudions les actions extensivement moyennables, une notion qui est un outil pour montrer la moyennabilité des groupes. Comme application, nous montrons la moyennabilité des groupes d'échanges d'intervalles dont les angles de translations ont rang rationnel au plus 2. Nous obtenons aussi une caractérisation "de type Kesten" de la moyennabilité extensive d'une action, et nous l'utilisons pour donner une preuve courte, purement probabiliste du fait que les actions récurrentes sont extensivement moyennables. Nous étudions aussi la propriété de Liouville pour les groupes d'échanges d'intervalles, et nous montrons qu'il existe des groupes d'échanges d'intervalles tels que toute mesure de support fini non dégénérée a un bord non trivial. Dans le Chapitre 5 (basé sur un travail en commun avec G. Amir, O. Angel, B. Virág) nous montrons que les groupes agissant sur les arbres enracinés par automorphismes bornés ont la propriété de Liouville. En particulier cela inclut les groupes engendrés par des automates d'activité bornée. / This thesis deals with the Liouville property and amenability of topological full groups of Cantor systems, groups of interval exchanges, and groups acting on rooted trees. In Chapter 2, we provide the first examples of finitely generated, infinite simple groups that have trivial Poisson-Furstenberg boundary for simple random walks (the Liouville property). These arise as the derived subgroup of the topological full groups of a family of minimal subshifts. We show that if the complexity of a (non necessarily minimal) subshift grows strictly subquadratically, every symmetric and finitely supported probability measure on the topological full group has vanishing asymptotic entropy. In Chapter 3, we exhibit a family of topological full groups of minimal subshifts that contain Grigorchuk groups G_ω as subgroups. This shows that the topological full group of a minimal subshift can have subgroups of intermediate growth, answering a question of Grigorchuk. In Chapter 4 (based on a joint work with K. Juschenko, N. Monod, M. de la Salle), we study various features of extensively amenable group actions, a notion which is a tool to prove amenability of groups. As an application, we prove amenability of groups of interval exchanges whose angular components have rational rank at most 2. We also obtain a "Kesten-like" characterisation of extensive amenability in terms of the inverted orbit and use it give a short, probabilistic proof of the fact that recurrent actions are extensively amenable. Finally we study the Liouville property for groups of interval exchanges, and show that there are groups of interval exchanges that admit no finitely supported measure with trivial boundary. In Chapter 5 (based on a joint work with G. Amir, O. Angel, B. Virág), we establish the Liouville property for all groups acting on rooted trees by bounded automorphisms. This includes in particular groups generated by bounded automata. This strengthens results by various authors about amenability of these groups, some of which are based on proving the Liouville property in some special cases.

Este trabalho visa a estudar o modelo de percolação independente, de Bernoulli, em grafos, tendo como objetivo principal obter condições que garantam a ocorrência de transição de fase. Iniciamos apresentando as definições e algumas técnicas fundamentais para o modelo de percolação (de elos ou de sítios) em um grafo infinito, conectado e localmente finito. Demonstramos então dois resultados essenciais: os fatos do parâmetro crítico não depender da escolha do vértice e da existência de um aglomerado infinito ter probabilidade 0 ou 1. Também obtemos um limitante inferior para o parâmetro crítico quando o grafo é de grau limitado. Para finalizar esta parte introdutória, analisamos a percolação em grafos particulares, a saber, a rede hipercúbica Z^d (para a qual mostramos a existência de transição de fase em dimensão d >= 2 e a unicidade do aglomerado infinito na fase supercrítica) e alguns tipos de árvores (para as quais apresentamos os parâmetros críticos). Na parte mais importante da dissertação, tendo como base os trabalhos de Benjamini e Schramm, de Häggström, Schonmann e Steif e de Lyons e Peres, introduzimos os conceitos de transitividade, amenabilidade e amenabilidade forte para um grafo. Fazemos uma detalhada discussão destas definições: provamos que a constante de Cheeger ancorada não depende do vértice em que é ancorada, estudamos relações entre os conceitos (amenabilidade e amenabilidade forte são noções distintas, bem como condições necessárias e suficientes para ambas) e calculamos a constante de Cheeger e a constante de Cheeger ancorada para alguns grafos. Finalmente, utilizando a técnica de crescimento do aglomerado, apresentamos para a probabilidade crítica um limitante superior que depende da constante ancorada. Isto nos permite concluir que ocorre transição de fase para qualquer grafo infinito, conectado, fracamente não-amenável (de constante de Cheeger ancorada positiva) e de grau limitado. / This work intends to study independent Bernoulli percolation model on graphs; the main purpose is obtaining conditions for phase transition. We begin presenting the definitions and some basic techniques for bond percolation and site percolation models on infinite, connected, locally finite graphs. We prove two essential results: the critical parameter is independent of the choice of the vertex and the probability that there exists an infinite cluster takes the values 0 and 1 only. We also obtain a lower bound for critical parameter when the graph is of bounded degree. To finish this preliminary part, we analyze percolation on particular graphs, namely the d-dimensional cubic lattice Z^d (for which we prove that there exists phase transition in dimension d >= 2 and the uniqueness of the infinite cluster in supercritical phase) and some trees (for which we present the critical parameters). In the most important part of this essay, founded in the works of Benjamini and Schramm, Häggström, Schonmann and Steif and Lyons and Peres, we introduce the concepts of transitivity, amenability and strong amenability. We discuss in detail these definitions: we prove that anchored Cheeger constant does not depend on the choice of the vertex, we study some relations (amenability and strong amenability are distinct notions, and necessary and sufficient conditions for both) and we obtain Cheeger constant and anchored Cheeger constant for some graphs. Finally, using the growing cluster technique, we present for the critical probability an upper bound that depends on the anchored constant. This permits us to conclude that there exists phase transition on infinite, connected, weakly non-amenable graphs of bounded degree.

Este trabalho visa a estudar o modelo de percolação independente, de Bernoulli, em grafos, tendo como objetivo principal obter condições que garantam a ocorrência de transição de fase. Iniciamos apresentando as definições e algumas técnicas fundamentais para o modelo de percolação (de elos ou de sítios) em um grafo infinito, conectado e localmente finito. Demonstramos então dois resultados essenciais: os fatos do parâmetro crítico não depender da escolha do vértice e da existência de um aglomerado infinito ter probabilidade 0 ou 1. Também obtemos um limitante inferior para o parâmetro crítico quando o grafo é de grau limitado. Para finalizar esta parte introdutória, analisamos a percolação em grafos particulares, a saber, a rede hipercúbica Z^d (para a qual mostramos a existência de transição de fase em dimensão d >= 2 e a unicidade do aglomerado infinito na fase supercrítica) e alguns tipos de árvores (para as quais apresentamos os parâmetros críticos). Na parte mais importante da dissertação, tendo como base os trabalhos de Benjamini e Schramm, de Häggström, Schonmann e Steif e de Lyons e Peres, introduzimos os conceitos de transitividade, amenabilidade e amenabilidade forte para um grafo. Fazemos uma detalhada discussão destas definições: provamos que a constante de Cheeger ancorada não depende do vértice em que é ancorada, estudamos relações entre os conceitos (amenabilidade e amenabilidade forte são noções distintas, bem como condições necessárias e suficientes para ambas) e calculamos a constante de Cheeger e a constante de Cheeger ancorada para alguns grafos. Finalmente, utilizando a técnica de crescimento do aglomerado, apresentamos para a probabilidade crítica um limitante superior que depende da constante ancorada. Isto nos permite concluir que ocorre transição de fase para qualquer grafo infinito, conectado, fracamente não-amenável (de constante de Cheeger ancorada positiva) e de grau limitado. / This work intends to study independent Bernoulli percolation model on graphs; the main purpose is obtaining conditions for phase transition. We begin presenting the definitions and some basic techniques for bond percolation and site percolation models on infinite, connected, locally finite graphs. We prove two essential results: the critical parameter is independent of the choice of the vertex and the probability that there exists an infinite cluster takes the values 0 and 1 only. We also obtain a lower bound for critical parameter when the graph is of bounded degree. To finish this preliminary part, we analyze percolation on particular graphs, namely the d-dimensional cubic lattice Z^d (for which we prove that there exists phase transition in dimension d >= 2 and the uniqueness of the infinite cluster in supercritical phase) and some trees (for which we present the critical parameters). In the most important part of this essay, founded in the works of Benjamini and Schramm, Häggström, Schonmann and Steif and Lyons and Peres, we introduce the concepts of transitivity, amenability and strong amenability. We discuss in detail these definitions: we prove that anchored Cheeger constant does not depend on the choice of the vertex, we study some relations (amenability and strong amenability are distinct notions, and necessary and sufficient conditions for both) and we obtain Cheeger constant and anchored Cheeger constant for some graphs. Finally, using the growing cluster technique, we present for the critical probability an upper bound that depends on the anchored constant. This permits us to conclude that there exists phase transition on infinite, connected, weakly non-amenable graphs of bounded degree.

Cette thèse porte sur l'étude des groupes polonais vus comme groupes d'automorphismes de structures métriques. L'observation que tout groupe polonais non archimédien est le groupe d'automorphismes d'une structure dénombrable ultra homogène a en effet mené à des interactions fructueuses entre la théorie des groupes et la théorie des modèles. Dans le cadre de la théorie des modèles métriques, introduite par Ben Yaacov, Henson et Usvyatsov, cette correspondance a été étendue par Melleray à tous les groupes polonais. Dans cette thèse, nous étudions diverses facettes de cette correspondance. Le lien entre une structure et son groupe d automorphismes est particulièrement étroit dans le cadre des structures ℵ0-categoriques. En effet, le théorème de reconstruction d'Ahlbrandt-Ziegler permet de retrouver une structure ℵ0-categorique, à bi-interprètabilité près, à partir de son groupe d'automorphismes. Dans un travail en commun avec Itai Ben Yaacov, nous généralisons ce résultat aux structures métriques separablement catégoriques. Les structures dénombrables ultra homogènes ont de plus l avantage d'être complètement déterminées par leurs sous-structures finiment engendrées. Cela a notamment permis a Moore de donner une caractérisation combinatoire de la moyennabilité des groupes polonais non archimédiens. Nous étendons cette caractérisation à tous les groupes polonais et nous en déduisons que la moyennabilite est une condition Gδ. Toujours dans une optique de reconstruction, nous nous intéressons à la propriété de continuité automatique pour les groupes polonais. Sabok et Malicki ont introduit des conditions de nature combinatoire sur une structure métrique ultra homogène qui impliquent la propriété de continuité automatique pour son groupe d'automorphismes. Nous montrons que ces conditions passent à la puissance dénombrable, ce qui a pour conséquence que les groupes Aut(μ)N, U(l2)N et Iso(U)N satisfont la propriété de continuité automatique. Ces conditions sont un affaiblissement du fait d'avoir des amples génériques. Dans un travail en commun avec Francois Le Maitre, nous exhibons les premiers exemples de groupes connexes qui ont des amples génériques, ce qui répond à une question de Kechris et Rosendal / This thesis focuses on the study of Polish groups seen as automorphism groups of metric structures. The observation that every non-archimedean Polish group is the automorphism group of an ultrahomogeneous countable structure has indeed led to fruitful interactions between group theory and model theory. In the framework of metric model theory, introduced by Ben Yaacov, Henson and Usvyastov, this correspondence has been extended to all Polish groups by Melleray. In this thesis, we study various facets of this correspondence. The relationship between a structure and its automorphism group is particularly close in the setting of ℵ0-categorical structures. Indeed, the Ahlbrandt-Ziegler reconstruction theorem allows one to recover an ℵ0-categorical structure, up to bi-interpretability, from its automorphism group. In a joint work with Itai Ben Yaacov, we generalize this result to separably categorical metric structures. Besides, ultrahomogeneous countable structures have the advantage of being completely determined by their finitely generated substructures. In particular, this enabled Moore to give a combinatorial characterization of amenability for nonarchimedean Polish groups. We extend this characterization to all Polish groups and we deduce that amenability is a Gδ condition. Still in a reconstruction perspective, we are interested in the automatic continuity property for Polish groups. Sabok and Malicki introduced conditions of a combinatorial nature on an ultrahomogeneous metric structure that imply the automatic continuity property for its automorphism group. We show that these conditions carry to countable powers, which leads to the groups Aut(μ)N, U(l2)N and Iso(U)N satisfying the automatic continuity property. Those conditions are a weakening of the property of having ample generics. In a joint work with Francois Le Maitre, we exhibit the first examples of connected groups with ample generics, which answers a question of Kechris and Rosendal. Finally, in a joint work with Isabel Muller and Aristotelis Panagiotopoulos, we study the relative homogeneity of substructures in an ultrahomogeneous countable structure. We characterize it completely by a property of the types over the substructures: being determined by a finite set

Dans cette thèse je m'intéresse à des propriétés de rigidité de certaines constructions d'algèbres de von Neumann. Ces constructions relient la théorie des groupes et la théorie ergodique au monde des algèbres d'opérateurs. Il est donc naturel de s'interroger sur la force de ce lien et sur la possibilité d'un enrichissement mutuel dans ces différents domaines. Le Chapitre II traite des actions Gaussiennes. Ce sont des actions de groupes discrets préservant une mesure de probabilité qui généralisent les actions de Bernoulli. Dans un premier temps, j'étudie les propriétés d'ergodicité de ces actions à partir d'une analyse de leurs algèbres de von Neumann (voir Theorem II.1.22 et Corollary II.2.16). Ensuite, je classifie les algèbres de von Neumann associées à certaines actions Gaussiennes, à isomorphisme près, en montrant un résultat de W*-Superrigidité (Theorem II.4.5). Ces résultats généralisent des travaux analogues sur les actions de Bernoulli ([KT08,CI10,Io11,IPV13]).Dans le Chapitre III, j'étudie les produits libres amalgamés d'algèbres de von Neumann. Ce chapitre résulte d'une collaboration avec C. Houdayer et S. Raum. Nous analysons les sous-Algèbres de Cartan de tels produits libres amalgamés. Nous déduisons notamment de notre analyse que le produit libre de deux algèbres de von Neumann n'est jamais obtenu à partir d'une action d'un groupe sur un espace mesuré.Enfin, le Chapitre IV porte sur les algèbres de von Neumann associées à des groupes hyperboliques. Ce chapitre est obtenu en collaboration avec A. Carderi. Nous utilisons la géométrie des groupes hyperboliques pour fournir de nouveaux exemples de sous-Algèbres maximales moyennables (mais de type I) dans des facteurs II_1. / The purpose of this dissertation is to put on light rigidity properties of several constructions of von Neumann algebras. These constructions relate group theory and ergodic theory to operator algebras.In Chapter II, we study von Neumann algebras associated with measure-Preserving actions of discrete groups: Gaussian actions. These actions are somehow a generalization of Bernoulli actions. We have two goals in this chapter. The first goal is to use the von Neumann algebra associated with an action as a tool to deduce properties of the initial action (see Corollary II.2.16). The second aim is to prove structural results and classification results for von Neumann algebras associated with Gaussian actions. The most striking rigidity result of the chapter is Theorem II.4.5, which states that in some cases the von Neumann algebra associated with a Gaussian action entirely remembers the action, up to conjugacy. Our results generalize similar results for Bernoulli actions ([KT08,CI10,Io11,IPV13]).In Chapter III, we study amalgamated free products of von Neumann algebras. The content of this chapter is obtained in collaboration with C. Houdayer and S. Raum. We investigate Cartan subalgebras in such amalgamated free products. In particular, we deduce that the free product of two von Neumann algebras is never obtained as a group-Measure space construction of a non-Singular action of a discrete countable group on a measured space.Finally, Chapter IV is concerned with von Neumann algebras associated with hyperbolic groups. The content of this chapter is obtained in collaboration with A. Carderi. We use the geometry of hyperbolic groups to provide new examples of maximal amenable (and yet type I) subalgebras in type II_1 factors.

Dans cette thèse, on s'intéresse à la théorie mesurée des groupes, à l'entropie sofique et aux algèbres d'opérateurs ; plus précisément, on étudie les actions des groupes sur des espaces de probabilités, des propriétés fondamentales de leur entropie sofique (pour des groupes discrets), leurs groupes pleins (pour des groupes Polonais), et les algèbres de von Neumann et leurs sous-algèbres moyennables (pour des groupes à caractère hyperbolique et des réseaux de groupes de Lie). Cette thèse est constituée de trois parties.Dans une première partie j'étudie l'entropie sofique des actions profinies. L'entropie sofique est un invariant des actions mesurées des groupes sofiques défini par L. Bowen qui généralise la notion d'entropie introduite par Kolmogorov. La définition d'entropie sofique nécessite de fixer une approximation sofique du groupe. Nous montrons que l'entropie sofique des actions profinies est effectivement dépendante de l'approximation sofique choisie dans le cas des groupes libres et certains réseaux de groupes de Lie.La deuxième partie est un travail en collaboration avec François Le Maître. Elle est constituée d'un article prépublié dans lequel nous généralisons la notion de groupe plein aux actions préservant une mesure de probabilité des groupes polonais, et en particulier, des groupes localement compacts. On définit une topologie polonaise sur ces groupes pleins et on étudie leurs propriétés topologiques fondamentales, notamment leur rang topologique et la densité des éléments apériodiques.La troisième partie est un travail en collaboration avec Rémi Boutonnet. Elle est constituée de deux articles prépubliés dans lesquels nous considérons la question de la maximalité de la sous-algèbre de von Neumann d'un sous-groupe moyennable maximal, dans celle du groupe ambiant. Nous résolvons la question dans le cas des groupes à caractère hyperbolique en utilisant les techniques de Sorin Popa. Puis, nous introduisons un critère dynamique à la Furstenberg, permettant de résoudre la question pour des sous-groupes moyennables de réseaux des groupes de Lie en rang supérieur. / This dissertation is about measured group theory, sofic entropy and operator algebras. More precisely, we will study actions of groups on probability spaces, some fundamental properties of their sofic entropy (for countable groups), their full groups (for Polish groups) and the amenable subalgebras of von Neumann algebras associated with hyperbolic groups and lattices of Lie groups. This dissertation is composed of three parts.The first part is devoted to the study of sofic entropy of profinite actions. Sofic entropy is an invariant for actions of sofic groups defined by L. Bowen that generalize Kolmogorov's entropy. The definition of sofic entropy makes use of a fixed sofic approximation of the group. We will show that the sofic entropy of profinite actions does depend on the chosen sofic approximation for free groups and some lattices of Lie groups. The second part is based on a joint work with François Le Maître. The content of this part is based on a prepublication in which we generalize the notion of full group to probability measure preserving actions of Polish groups, and in particular, of locally compact groups. We define a Polish topology on these full groups and we study their basic topological properties, such as the topological rank and the density of aperiodic elements. The third part is based on a joint work with Rémi Boutonnet. The content of this part is based on two prepublications in which we try to understand when the von Neumann algebra of a maximal amenable subgroup of a countable group is itself maximal amenable. We solve the question for hyperbolic and relatively hyperbolic groups using techniques due to Popa. With different techniques, we will then present a dynamical criterion which allow us to answer the question for some amenable subgroups of lattices of Lie groups of higher rank.

We extend Følner’s amenability criterion to the realm of general topological groups. Building on this, we show that a topological group G is amenable if and only if its left-translation action can be approximated in a uniform manner by amenable actions on the set G. As applications we obtain a topological version of Whyte’s geometric solution to the von Neumann problem and give an affirmative answer to a question posed by Rosendal.