<b>Structural Results for the Von Neumann Algebras of Thompson-Like Groups</b>

<p dir="ltr">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.</p><p dir="ltr">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.</p><p dir="ltr">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.</p><p dir="ltr">Together, these results exhibit the breadth of properties possible for the von Neumann</p>