On full groups of non ergodic probability measure preserving equivalence relations

François Le Maître (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We give a formula relating the topological rank of the full group of an aperiodic pmp equivalence relation to the cost of its ergodic components. Furthermore, we obtain examples of full groups having a dense free subgroup whose rank is equal to the topological rank of the full group, using a Baire category argument. We then study the automatic continuity property for full groups of aperiodic equivalence relations, and find a connected metric for which they have the automatic continuity property. This allows us to give an algebraic characterization of aperiodicity for pmp equivalence relations, namely the non-existence of homomorphisms from their full groups into totally disconnected separable groups. A simple proof of the extreme amenability of full groups of hyperfinite pmp equivalence relations is also given, generalizing to the non ergodic case a result of Giordano and Pestov.

Mots clés: pmp equivalence relations; full groups; topological groups

Domaines: Systèmes dynamiques [math.DS]; Théorie des groupes [math.GR]

François Le Maître. On full groups of non ergodic probability measure preserving equivalence relations. Ergodic Theory and Dynamical Systems, 2015, 36 (7), pp.2218-2245. ⟨10.1017/etds.2015.20⟩. ⟨hal-00992639v2⟩