Structural Results for the Von Neumann Algebras of Thompson-Like Groups

When considering von Neumann algebras, a II_1 factor is said to be prime if it does not decompose as the tensor product of two II_1 factors. Using Popa’s deformation/rigidity theory, we prove that the Higman-Thompson groups Td and Vd give rise to prime group von Neumann algebras. Part of the novelty of this result is the use of deformations associated to a quasi-regular representation not weakly contained in the left regular representation. In fact, a much larger class of groups satisfy this primeness result; a certain family of generalized Thompson groups that arise as groups of local similarities of a compact ultrametric space known as CSS groups, a natural extension of Hughes’ FSS groups. This application required substantial new group theory and in the process, we proved that a subclass of CSS groups, referred to as CSS* groups, are non-inner amenable and properly proximal.

While the class of non-inner amenable group is quite large, there has been particular attention to proving that generalized Thompson groups are non-inner amenable, as they were not known to fall into a class of non-inner amenable groups. In particular, our result proves that the Röver-Nekrashevych groups are non-inner amenable, answering a question of Bashwinger and Zaremsky. Properly Proximal groups are a relatively new class of non-inner amenable groups. They generalize bi-exactness and possess actions that produce von Neumann algebras with desirable properties. Thus, it is of great interest to know which groups are properly proximal.

In another direction, we proved that a large class of Thompson-Like groups arising from the d-ary cloning-system construction of Skipper and Zaremsky give rise to a stable orbit equivalence relation. Consequently, they are McDuff groups, meaning that they admit a free ergodic p.m.p action on a probability space, such that the resulting crossed-product von Neumann algebra is McDuff.

Together, these results exhibit the breadth of properties possible for the von Neumann