McDuff factors from amenable actions and dynamical alternating groups

David Kerr¹ ; Spyridon Petrakos¹ 

¹University of Münster, Germany

Groups, geometry, and dynamics, Tome 19 (2025) no. 2, pp. 415-429

Résumé

Given a topologically free action of a countably infinite amenable group on the Cantor set, we prove that, for every subgroup G of the topological full group containing the alternating group, the group von Neumann algebra LG is a McDuff factor. This yields the first examples of nonamenable simple finitely generated groups G for which LG is McDuff. Using the same construction we show moreover that if a faithful action G↷X of a countable group on a countable set with no finite orbits is amenable then the crossed product of the associated shift action over a given II1​ factor is a McDuff factor. In particular, if H is a nontrivial countable ICC group and G↷X is a faithful amenable action of a countable ICC group on a countable set with no finite orbits, then the group von Neumann algebra of the generalized wreath product H≀X​G is a McDuff factor. Our technique can also be applied to show that if H is a nontrivial countable group and G↷X is an amenable action of a countable group on a countable set with no finite orbits, then the generalized wreath product H≀X​G is Jones–Schmidt stable.

Détail

DOI : 10.4171/ggd/880

Classification : 22D25
Mots-clés : II1​ factor, McDuff property, topological full group