A connected, locally finite graph $\Gamma $ is a Cayley–Abels graph for a totally disconnected, locally compact group G if G acts vertex-transitively on $\Gamma $ with compact, open vertex stabilizers. Define the minimal degree of G as the minimal degree of a Cayley–Abels graph of G. We relate the minimal degree in various ways to the modular function, the scale function and the structure of compact open subgroups. As an application, we prove that if $T_{d}$ denotes the d-regular tree, then the minimal degree of $\mathrm{Aut}(T_{d})$ is d for all $d\geq 2$ .

This thesis is at the intersection of dynamics, probability and model theory. It focuses on a specialization of the notion of amenability: unique ergodicity.

Let G be a Polish group, i.e., a topological group whose topology is separable and completely metrizable. We call a G-flow the action of G on a compact space. A G-flow is said to be minimal if every orbit is dense.

A famous theorem of Ellis states that any Polish group G admits a unique universal minimal flow that we denote ${\mathrm {M}}(G)$ . This means that for any minimal G-flow X there is a surjective G-map from ${\mathrm {M}}(G)$ to X. G is said to be amenable if every G-flow admits an invariant probability measure, and uniquely ergodic if every minimal flow admits a unique invariant probability measure.

The notion of unique ergodicity relating to a group was introduced by Angel, Kechris and Lyons. They also ask the following question which is the main focus of the thesis: Let G be an amenable Polish group with metrizable universal minimal flow, is G uniquely ergodic?

Note that unique ergodicity is an interesting notion only for relatively large groups, as it is proved in the last chapter of this thesis that locally compact non compact Polish groups are never uniquely ergodic. This result is joint work with Andy Zucker.

The thesis includes proofs of unique ergodicity of groups with interesting universal minimal flows, namely the automorphism group of the semigeneric directed graph and the automorphism group of the $2$ -graph.

It also includes a theorem stating that under some hypothesis on a $\omega $ -categorical structure M, the logic action of ${\mathrm {Aut}}(M)$ on ${\mathrm {LO}}(M)$ , the compact space of linear orders on M, is uniquely ergodic. This implies unique ergodicity for the group if its universal minimal flow happens to be the space of linear orderings. It can also be used to prove non-amenability of some groups for which the action of ${\mathrm {Aut}}(M)$ on ${\mathrm {LO}}(M)$ is not minimal. This result is joint work with Todor Tsankov.

Finally, the thesis also presents a proof that under the assumption that the universal minimal flows involved are metrizable, unique ergodicity is stable under group extensions. This result is joint work with Andy Zucker.

Abstract prepared by Colin Jahel.

E-mail: [email protected]

URL: http://math.univ-lyon1.fr/~jahel/doc/These.pdf

We prove that the automorphism group of the semigeneric directed graph (in the sense of Cherlin’s classification) is uniquely ergodic.

Given a G-flow X, let $\mathrm{Aut}(G, X)$ , or simply $\mathrm{Aut}(X)$ , denote the group of homeomorphisms of X which commute with the G action. We show that for any pair of countable groups G and H with G infinite, there is a minimal, free, Cantor G-flow X so that H embeds into $\mathrm{Aut}(X)$ . This generalizes results of [2, 7].

Let $\unicode[STIX]{x1D6F4}$ be a compact orientable surface of genus $g=1$ with $n=1$ boundary component. The mapping class group $\unicode[STIX]{x1D6E4}$ of $\unicode[STIX]{x1D6F4}$ acts on the $\mathsf{SU}(3)$-character variety of $\unicode[STIX]{x1D6F4}$. We show that the action is ergodic with respect to the natural symplectic measure on the character variety.

It is a well-known fact that the greatest ambit for a topological group $G$ is the Samuel compactification of $G$ with respect to the right uniformity on $G$. We apply the original description by Samuel from 1948 to give a simple computation of the universal minimal flow for groups of automorphisms of uncountable structures using Fraϊssé theory and Ramsey theory. This work generalizes some of the known results about countable structures

A topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann algebra is approximately finite dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group, which strengthens a result by de la Harpe. As a consequence, a $C^\ast$-algebra $A$ is nuclear if and only if the unitary group $U(A)$ with the relative weak topology is strongly amenable in the sense of Glasner. We prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology and establish a similar result for groups of non-singular transformations. As a consequence, we prove extreme amenability of the groups of isometries of $L^p(0,1)$, $1\leq p<\infty$, extending a classical result of Gromov and Milman ($p=2$). We show that a measure class preserving equivalence relation $\mathcal{R}$ on a standard Borel space is amenable if and only if the full group $[\mathcal{R}]$, equipped with the uniform topology, is extremely amenable. Finally, we give natural examples of concentration to a non-trivial space in the sense of Gromov occurring in the automorphism groups of injective factors of type III.

Necessary and sufficient conditions are given for a Polish topological group to be ‘almost free’. It is deduced that the existence of one free subgroup of a Polish group can lead to the existence of many free subgroups of maximal rank. Applications are given to permutation groups, profinite groups, Lie groups and unitary groups.