Abstract:Countable Similarity Structure (CSS) groups are a class of generalized Thompson groups essentially introduced by Hughes. In this paper, we study CSS$^*$ groups, a subclass that includes the Higman-Thompson groups $V_{d,r}$, the countable Röver-Nekrashevych groups $V_d(G)$, and the topological full groups of subshifts of finite type of Matui. We prove that many CSS$^*$ groups give rise to prime group von Neumann algebras, greatly expanding the class of groups satisfying the result of the second named author, de Santiago, and Khan. In the process, we also prove that many CSS$^*$ groups are non-inner amenable and properly proximal. We then prove 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. Lastly, we observe that CSS$^*$ groups are not acylindrically hyperbolic, motivating the need to prove many of these results by other methods.
| Comments: | 50 pages, 3 figures. v2: Section 6 rewritten and simplified. Reviewer comments and suggestions implement. Minor modifications throughout. To appear in Algebraic & Geometric Topology |
| Subjects: | Operator Algebras (math.OA); Group Theory (math.GR) |
| MSC classes: | 46L36, 46L35 (Primary), 20F65, 20E32 (Secondary) |
| Cite as: | arXiv:2507.18821 [math.OA] |
| (or arXiv:2507.18821v2 [math.OA] for this version) | |
| https://doi.org/10.48550/arXiv.2507.18821 arXiv-issued DOI via DataCite |
From: Patrick DeBonis [view email]
[v1]
Thu, 24 Jul 2025 21:43:47 UTC (53 KB)
[v2]
Fri, 22 May 2026 15:12:01 UTC (51 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。