Ergodic and Geometric Group Theory
École polytechnique fédérale de Lausanne
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maxime.gheysens AD epfl.ch
We investigate how probability tools can be useful to study representations of nonamenable groups. A suitable notion of 'probabilistic subgroup' is proposed for locally compact groups, and is valuable to induction of representations. Nonamenable groups admit nonabelian free subgroups in that measure-theoretical sense. Consequences for affine actions and for unitarizability are then drawn. In particular, we obtain a new characterization of amenability via some affine actions on Hilbert spaces.
Along the way, various fixed-point properties for groups are studied. We also give a survey of several useful facts about group representations on Banach spaces, continuity of group actions, compactness of convex hulls in locally convex spaces, and measurability pathologies in Banach spaces.
Keywords: amenability, group representation, fixed-point property, von Neumann problem, Dixmier problem, induced representation, tychomorphism, Krein space.
This exposition article arose from two talks given during the Oberwolfach Arbeitsgemeinschaft on Totally Disconnected Groups in October 2014.
This is an introduction to the structure theory of totally disconnected locally compact groups initiated by Willis in 1994. The two main tools in this theory are the scale function and tidy subgroups, for which we present several properties and examples.
As an illustration of this theory, we give a proof of the fact that the set of periodic elements in a totally disconnected locally compact group is always closed, and that such a group cannot have ergodic automorphisms as soon as it is non-compact.
Consider the following property of a topological group G: every continuous affine G-action on a Hilbert space with a bounded orbit has a fixed point. We prove that this property characterizes amenability for locally compact σ-compact groups (e.g. countable groups).
Along the way, we introduce a "moderate" variant of the classical induction of representations and we generalize the Gaboriau–Lyons theorem to prove that any non-amenable locally compact group admits a probabilistic variant of discrete free subgroups. This leads to the "measure-theoretic solution" to the von Neumann problem for locally compact groups.
We illustrate the latter result by giving a partial answer to the Dixmier problem for locally compact groups.
Assistanat pour les cours suivants :
Encadrement d'étudiants pour divers projets de semestre (voir la page des projets de semestre de la chaire EGG).
Édition du rapport de l'Arbeitsgemeinschaft Totally disconnected groups d'Oberwolfach (octobre 2014).
Organisation d'un groupe de travail sur la structure des groupes simples totalement discontinus (printemps 2014).
Organisation du quatorzième colloque des doctorants en mathématique de Suisse occidentale (février 2015).
Délégué des doctorants mathématiciens de l'EPFL auprès du Programme doctoral suisse en mathématiques (depuis mars 2014).