Une structure dénombrable du premier ordre est dite homogène si tout isomorphisme entre deux sous-Structures finiment engendrées s’étend en un automorphisme de la structure globale.C’est équivalent à une propriété d’amalgamation des sous-Structures finiment engendrées, et les structures homogènes dénombrables sont aussi appelées limites de Fraïssé, en lien avec les travaux de Roland Fraïssé sur l’ordre des rationnels. Cette thèse concerne les groupes d’automorphismesdes structures homogènes, avec la question centrale suivante: est-Ce que le groupe automorphismes d’une structure homogène est universel pour la classe des groupes d’automorphismes de ces sous-Structures ? Nous répondons positivement à cette question pour les structures homogènesdans un langage relationnel et avec la propriété d’amalgamation libre, à l’aide d’une construction par tour assez similaire à une construction de Katetov et Uspenskij dans le cas de l’espace d’Urysohn. Avec des techniques similaires, nous obtenons toute sous-Structure dénombrable comme points fixes d’un automorphisme d’ordre fini pré-Déterminé. Cela nous permet par ailleurs d’étudier la complexité de la relation d’isomorphisme entre sous-Structures dénombrables, et de montrer qu’elle se réduit boreliennement à la relation de conjugaison dans le groupe d’automorphismes. Nous continuons avec les éléments d’ordre fini, en supposant de plus que les sous-Structures finies satisfont une version forte de la propriété d’extension de Hrushovski-Lascar-Herwig, et des arguments topologiques nous permettent alors de montrer que dans le groupe d’automorphismes tout élément est produit de quatre conjugués de certains éléments d’ordre fini. Nous montrons aussi des résultats similaires pour le groupe d’isométries de l’espace d’Urysohn,ou sa version bornée, la sphère d’Urysohn, en utilisant le fait que ces derniers sont très bien approximés par des espaces métriques rationnels. Enfin, revenant à la question de l’universalité du groupe automorphismes de la limite de Fraïssé, nous considérons la question plus fine de savoirsi toute sous-Structure dénombrable s’injecte de manière rigide, c’est-À-Dire de sorte chacun de ces automorphismes s’étende en un unique automorphisme de la limite de Fraïssé. D’abord, nous introduisons une construction de telle injections rigides dans le cas des graphes homogènes. Ensuite, nous modifions cette construction dans diverses classes de graphes orientés et de structures relationnelles homogènes, pour enfin la faire fonctionner dans un contexte très general de structures dans un langage relationnel fini et avec la propriété d’amalgamation libre. / A countable first-Order structure is called homogneous when each isomorphism between twofinitely generated substructures extends to an automorphism of the whole structure. This is equivalentto an amalgamation property of finitely generated substructures, and countable homogeneousstructures are also called Fraïssé limits, in connection to the work of Roland Fraïssé on theorder of rational numbers. The present thesis concerns automorphism groups of homogeneousstructures, with the following central question: is it the case that the automorphism group of a homogeneousstructure is universal for the class of automorphism groups of its substructures? Weanswer positively this question for homogeneous structures in a relational langage and with thefree amalgamation property, by using a construction rather similar to a construction of Katetov andUspenskij in the case of the Urysohn space.With similar techniques, we obtain any countable substructureas the set of fixed points of an automorphism of a given finite order. Besides, this allowsus to study the complexity of the isomorphism relation between countable substructures, and toshow that it Borel reduces to the conjugacy relation in the automorphism group. We continue withelements of finite order, assuming further that finite substructures satisfy a strong version of theHrushovski-Lascar-Herwig extension property, and topological arguments then allow us to showthat in the automorphism group any element is the product of four conjugates of certain elementsof finite order. We also show similar results for the isometry group of the Urysohn space, or itsbounded version, the Urysohn sphere, by using the fact that they are well approximated by rationalmetric spaces. Finally, concerning the question of the universality of the automorphism groupof a Fraïssé limit, we consider the finer question to know whether any countable substructure embedsin a rigid way, that is, in such a way that each of its automorphisms extends in a uniqueautomorphism of the Fraïssé limit. First, we introduce a construction of such rigid embeddings inthe case of homogeneous graphs. Then, we modify this construction in various classes of orientedgraphs and of homogeneous relational structures, ultimately to make it work in a very generalcontext of structures in a finite relational langage and with the free amalgamation property.

In this thesis, we present a new viewpoint of the universal minimal flow in the language of near ultrafilters. We apply this viewpoint to generalize results of Kechris, Pestov and Todorcevic about a connection between groups of automorphisms of structures and structural Ramsey theory from countable to uncountable structures. This allows us to provide new examples of explicit descriptions of universal minimal flows as well as of extremely amenable groups. We identify new classes of finite structures satisfying the Ramsey property and apply the result to the computation of the universal minimal flow of the group of automorphisms of $\P(\omega_1)/\fin$ as well as of certain closed subgroups of groups of homeomorphisms of Cantor cubes. We furthermore apply our theory to groups of isometries of metric spaces and the problem of unique amenability of topological groups. The theory combines tools from set theory, model theory, Ramsey theory, topological dynamics and ergodic theory, and homogeneous structures.

In this thesis, we present a new viewpoint of the universal minimal flow in the language of near ultrafilters. We apply this viewpoint to generalize results of Kechris, Pestov and Todorcevic about a connection between groups of automorphisms of structures and structural Ramsey theory from countable to uncountable structures. This allows us to provide new examples of explicit descriptions of universal minimal flows as well as of extremely amenable groups. We identify new classes of finite structures satisfying the Ramsey property and apply the result to the computation of the universal minimal flow of the group of automorphisms of $\P(\omega_1)/\fin$ as well as of certain closed subgroups of groups of homeomorphisms of Cantor cubes. We furthermore apply our theory to groups of isometries of metric spaces and the problem of unique amenability of topological groups. The theory combines tools from set theory, model theory, Ramsey theory, topological dynamics and ergodic theory, and homogeneous structures.

This work analyses properties of relational structures that imply a high degree of symmetry. A structure is called homogeneous if every mapping from any finite substructure can be extended to a mapping over the whole structure. The various types of these mappings determine corresponding types of homogeneity. A prominent position belongs to ultrahomogeneity, for which every local isomorphism can be extended to an automorphism. In contrast to graphs, the classification of ultrahomogeneous relational struc- tures is still an open problem. The task of this work is to characterize "the distance" to homogeneity using two approaches. Firstly, the classification of homogeneous structures is studied when the "complexity" of a structure is increased by introducing more relations. This leads to various classifications of homomorphism-homogeneous L-colored graphs for different L, where L- colored graphs are graphs having sets of colors from a partially ordered set L assigned to vertices and edges. Moreover a hierarchy of classes of ho- mogeneous structures defined via types of homogeneity is studied from the viewpoint of classes coincidence. The second approach analyses for fixed classes of structures the least way to extend their language so as to achieve homogeneity. We obtain results about relational complexity for finite...