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Eli Bashwinger

Publications and source records attributed to Eli Bashwinger.

4 recordsLinked to original sources
arXiv2026

On Similarity Structure Groups and their W$^*$ and C$^*$-Algebras

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.

math.OA↗
arXiv2025

Non-inner amenability of the Higman-Thompson groups

We prove that the Higman-Thompson groups $T_n$ and $V_n$ are non-inner amenable for all $n\ge 2$. This extends Haagerup and Olesen's result that Thompson's groups $T=T_2$ and $V=V_2$ are non-inner amenable. Their proof relied on machinery only available in the $n=2$ case, namely Thurston's piecewise-projective model for Thompson's group $T$, so our approach necessarily utilizes different tools. This also provides an alternate proof of Haagerup-Olesen's result when $n=2$.

math.GR↗
arXiv2024

Von Neumann Algebras of Thompson-like Groups from Cloning Systems II

Let $(G_n)_{n \in \mathbb{N}}$ be a sequence of groups equipped with a $d$-ary cloning system and denote by $\mathscr{T}_d(G_*)$ the resulting Thompson-like group. In previous work joint with Zaremsky, we obtained structural results concerning the group von Neumann algebra of $\mathscr{T}_d(G_*)$, denoted by $L(\mathscr{T}_d(G_*))$. Under some natural assumptions on the $d$-ary cloning system, we proved that $L(\mathscr{T}_d(G_*))$ is a type $\text{II}_1$ factor. With a few additional natural assumptions, we proved that $L(\mathscr{T}_d(G_*))$ is, moreover, a McDuff factor. In this paper, we further analyze the structure of $L(\mathscr{T}_d(G_*))$, in particular the inclusion $L(F_d) \subseteq L(\mathscr{T}_d(G_*))$, where $F_d$ is the smallest of the Higman--Thompson groups. We prove that if the $d$-ary cloning system is ``diverse," then $L(F_d) \subseteq L(\mathscr{T}_d(G_*))$ satisfies the weak asymptotic homomorphism property. As a consequence, the inclusion is irreducible, which is a considerable improvement of our result that $L(\mathscr{T}_d(G_*))$ is a type $\text{II}_1$ factor, and the inclusion is also singular. Then we look at examples of non-diverse $d$-ary cloning systems with respect to the weak asymptotic homomorphism property, singularity, and irreducibility. Then we finish the paper with some applications. We construct a machine which takes in an arbitrary group and finite group and produces an inclusion (both finite and infinite index) of type $\text{II}_1$ factors which is singular but without the weak asymptotic homomorphism property. Finally, using irreducibility of the inclusion $L(F_d) \subseteq L(\mathscr{T}_d(G_*))$, our conditions for when $L(\mathscr{T}_d(G_*))$ is a McDuff factor, and the fact that Higman-Thompson groups $F_d$ are character rigid (in the sense of Peterson), we prove that the groups $F_d$ are McDuff (in the sense of Deprez-Vaes).

math.OA↗
arXiv2022

Von Neumann algebras of Thompson-like groups from cloning systems

We prove a variety of results about the group von Neumann algebras associated to Thompson-like groups arising from so called $d$-ary cloning systems. Cloning systems are a framework developed by Witzel and the second author, with a $d$-ary version subsequently developed by Skipper and the second author, which can be used to construct generalizations of the classical Thompson's groups $F$, $T$, and $V$. Given a family of groups $(G_n)_{n\in\mathbb{N}}$ with a $d$-ary cloning system, we get a Thompson-like group $\mathscr{T}_d(G_*)$, and in this paper we find some mild, natural conditions under which the group von Neumann algebra $\mathcal{L}(\mathscr{T}_d(G_*))$ has desirable properties. For instance, if the $d$-ary cloning system is "fully compatible" and "diverse" then we prove that $\mathcal{L}(\mathscr{T}_d(G_*))$ is a type $\textrm{II}_1$ factor. If moreover the $d$-ary cloning system is "uniform" and "slightly pure" then we prove $\mathcal{L}(\mathscr{T}_d(G_*))$ is even a McDuff factor, so $\mathscr{T}_d(G_*)$ is inner amenable. Examples of $d$-ary cloning systems satisfying these conditions are easy to come by, and include many existing examples, for instance our results show that for $bV$ and $bF$ the Brin-Dehornoy braided Thompson group and pure braided Thompson group, $\mathcal{L}(bV)$ and $\mathcal{L}(bF)$ are type $\textrm{II}_1$ factors and $\mathcal{L}(bF)$ is McDuff. In particular we get the surprising result that $bF$ is inner amenable.

math.OA↗