McDuff factors from amenable actions and dynamical alternating groups

Abstract

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 $\mathcal{L}G$ is a McDuff factor. This yields the first examples of nonamenable simple finitely generated groups $G$ for which $\mathcal{L}G$ is McDuff. Using the same construction we show moreover that if a faithful action $G\curvearrowright X$ 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 II${}_1$ factor is a McDuff factor. In particular, if $H$ is a nontrivial countable ICC group and $G\curvearrowright X$ 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 $H{\wr }_{X}G$ is a McDuff factor. Our technique can also be applied to show that if $H$ is a nontrivial countable group and $G\curvearrowright X$ is an amenable action of a countable group on a countable set with no finite orbits, then the generalized wreath product $H{\wr }_{X}G$ is Jones–Schmidt stable.