McDuff and Prime von Neumann algebras arising from Thompson-Like Groups

In this paper we show that the cloning system construction of Skipper and Zaremsky [SZ21], under sufficient conditions, gives rise to Thompson-Like groups which are stable; in particular, these are McDuff groups in the sense of Deprez and Vaes [DV18]. This answers a question of Bashwinger and Zaremsky posed in [BZ23] in the affirmative. In the opposite direction, we show that the group von Neumann algebra for the Higman-Thompson groups $T_d$ and $V_d$ are both prime II${}_1$ factors. This follows from a new deformation/rigidity argument for a certain class of groups which admit a proper cocycle into a quasi-regular representation that is not necessarily weakly ${\mathrm{\ell }}^2$.