Title: The W* and C*-algebras of Similarity Structure Groups

Abstract: Countable Similarity Structure (CSS) groups are a class of generalized Thompson groups. I will introduce CSS* groups, a subclass, that we prove to be non-acylindrically hyperbolic, that includes the Higman-Thompson groups Vd,r , the countable Röver-Nekrashevych groups Vd(G), and the topological full groups of subshifts of finite type of Matui. I will discuss how all CSS* groups give rise to prime group von Neumann algebras, which greatly expands the class of groups satisfying a previous deformation/rigidity result. I will then discuss how 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. This is joint work with Eli Bashwinger.