McDuff factors from amenable actions and dynamical alternating groups

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 II ₁ 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 HX 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 HX G is Jones–Schmidt stable.