Sur les groupes pleins préservant une mesure

THÈSE
en vue de l’obtention du grade de
Docteur de l’Université de Lyon
délivré par l’École Normale Supérieure de Lyon
Discipline : Mathématiques
Unité de Mathématiques Pures et Appliquées
École doctorale InfoMaths
présentée et soutenue publiquement le 12 mai 2014
par M. François Le Maître
Sur les groupes pleins préservant une
mesure de probabilité
Directeurs de thèse : M. Damien Gaboriau
M. Julien Melleray
Après avis de : M. Thierry Giordano
M. Alexander S. Kechris
Devant la commission d’examen formée de :
M. Damien Gaboriau, ENS de Lyon, Directeur
M. Thierry Giordano, University of Ottawa, Rapporteur
M. Cyril Houdayer, ENS de Lyon, Examinateur
M. Gilbert Levitt, Université de Caen, Examinateur
M. Alain Louveau, Université Paris 6, Examinateur
M. Julien Melleray, Université Lyon 1, Directeur
M. Todor Tsankov, Université Paris 7, Examinateur

à FHADXE.

Abstract
Let (X, µ)be a standard probability space and Γa countable group acting on Xin
a measure preserving way. The partition of the space Xinto Γ-orbits is entirely encoded
by the full group of the action, consisting of all the Borel bijections of Xwhich act by
permutation on every orbit. To be more precise, Dye’s reconstruction theorem states that
two measure preserving actions are orbit equivalent (i.e. they induce the same partition up
to a measure preserving bijection of (X, µ)) if and only if their full groups are isomorphic.
The reconstruction theorem is the main motivation for this thesis, in which we try
to understand how exactly orbit equivalence invariants of measure preserving actions
translate into algebraic or topological properties of the associated full group.
The main result deals with the topological rank of full groups, that is the minimal
number of elements needed to generate a dense subgroup. It happens to be deeply linked
to a fundamental invariant of orbit equivalence : the cost. To be more precise, we have
shown that the topological rank is, in the ergodic case, equal to the integer part of the
cost of the action plus one. The non-ergodic case was also treated, and we obtained some
genericity results for the set of topological generators.
We also obtained a characterization of the measure preserving actions having only
infinite orbits : these are the ones whose full group has no nontrivial morphism into
Z/2Z.