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We study lattice embeddings for the class of countable groups $\Gamma$ defined by the property that the largest amenable uniformly recurrent subgroup $A_\Gamma$ is continuous. When $A_\Gamma$ comes from an extremely proximal action and the envelope of $A_\Gamma$ is co-amenable in $\Gamma$, we obtain restrictions on the locally compact groups $G$ that contain a copy of $\Gamma$ as a lattice, notably regarding normal subgroups of $G$, product decompositions of $G$, and more generally dense mappings from $G$ to a product of locally compact groups., Comment: v1: 44 pages, preliminary version. v2: slightly modified version. v3: modified terminology, added paragraph 6.5.4. v4: Part of Section 6 has been extracted to arXiv:2001.08689

This thesis is at the intersection of dynamics, combinatorics and probability theory. My work focuses on a specialization of the notion of amenability: unique ergodicity. This notion refers to those groups whose minimal actions admit a unique invariant measure. I am especially interested in automorphism groups of Fraïssé limits. Those groups have several interesting properties. One of those properties is that they are somewhat "big", meaning not locally compact. The case of locally compact groups is discussed in Chapter 6. Another interesting property of automorphism groups is the understanding we have of some of their actions. This is deeply linked to the fact that we are dealing with automorphism groups. My work essentially relies on the understanding of those actions to construct measures that allow us to understand all the invariant measures of soThis thesis is at the intersection of dynamics, combinatorics and probability theory. My work focuses on a specialization of the notion of amenability: unique ergodicity. This notion refers to those groups whose minimal actions admit a unique invariant measure. I am especially interested in automorphism groups of Fraïssé limits. Those groups have several interesting properties. One of those properties is that they are somewhat "big", meaning not locally compact. The case of locally compact groups is discussed in Chapter 6. Another interesting property of automorphism groups is the understanding we have of some of their actions. This is deeply linked to the fact that we are dealing with automorphism groups. My work essentially relies on the understanding of those actions to construct measures that allow us to understand all the invariant measures of some groups., Cette thèse est à l’intersection de la dynamique, de la combinatoire et de la théorie des probabilités. Mon travail se concentre sur une spécialisation de la notion de moyennabil-ité : l’unique ergodicité. Il s’agit d’une qualification pour décrire les groupes pour lesquelles les actions minimales admettent exactement une mesure invariante. Je m’intéresse partic-ulièrement à des groupes d’automorphismes de limites de Fraïssé qui ont cette propriété. Les groupes d’automorphismes de limites de Fraïssé ont plusieurs propriétés qui les rendent très intéressants. Premièrement, il sont assez "gros", c’est à dire non localement compact. Le chapitre 6 traite le cas des groupes localement compacts. Une autre propriété intéressante est la compréhension que nous avons de la topologie de certains espaces sur lesquels agissent ces groupes, ceci étant directement lié au fait qu’on a à faire à des groupes d’automorphismes. Mon travail consiste principalement à exploiter notre compréhension de ces actions pour con-struire des mesures qui nous permettent de comprendre toutes les mesures invariantes pour certains groupes.