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 $\ell^2$.
Comment: Groups action on trees removed. Fix the proof Theorem 4.5 which is now Theorem 4.10. Removed an assumption from Theorem A (Theorem 3.3). Fixed the proof of Theorem D/corollary 6.5 (now Theorem C/corollary 5.11). Removed Theorem C. Fix Theorem 4.5 (now Theorem 4.10). Combined section 5 and 6. Corrected the proof of lemma 6.1 (now lemma 5.6). Corrected the proof of lemma 6.1 (now lemma 5.6)