On Similarity Structure Groups and their W$^*$ and C$^*$-Algebras

Abstract

Countable Similarity Structure (CSS) groups are a class of generalized Thompson groups essentially introduced by Hughes. In this paper, we study CSS$^*$ groups, a subclass that includes the Higman-Thompson groups $V_{d,r}$, the countable R\"over-Nekrashevych groups $V_d(G)$, and the topological full groups of subshifts of finite type of Matui. We prove that many CSS$^*$ groups give rise to prime group von Neumann algebras, greatly expanding the class of groups satisfying the result of the second named author, de Santiago, and Khan. In the process, we also prove that many CSS$^*$ groups are non-inner amenable and properly proximal. We then prove CSS$^*$ groups are either $C^*$-simple with a simple commutator subgroup, or lack both properties. This extends $C^*$-simplicity results of Le Boudec and Matte Bon and recovers the simple commutator subgroup results of Bleak, Elliott, and Hyde. Lastly, we observe that CSS$^*$ groups are not acylindrically hyperbolic, motivating the need to prove many of these results by other methods.