Date of Award
5-1-2024
Language
English
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
College/School/Department
Department of Mathematics and Statistics
Dissertation/Thesis Chair
Matthew Zaremsky
Subject Categories
Physical Sciences and Mathematics
Abstract
Let $(G_n)_{n \in \mathbb{N}}$ be a sequence of groups equipped with a so-called $d$-ary cloning system, and denote by $\mathscr{T}_d(G_*)$ the resulting Thompson-like group, which ought to be regarded as a ``Thompson-esque'' limit of the sequence $(G_n)_{n \in \mathbb{N}}$. In this thesis, we record the progress made so far in our studies on the von Neumann algebras of Thompson-like groups arising from $d$-ary cloning systems. For example, we study the inclusion $L(F_d) \subseteq L(\mathscr{T}_d(G_*))$ of group von Neumann algebras, where $F_d$ is the smallest of the Higman-Thompson groups. We prove that if $(G_n)_{n \in \mathbb{N}}$ is equipped with a ``diverse" $d$-ary cloning system, then the inclusion $L(F_d) \subseteq L(\mathscr{T}_d(G_*))$ satisfies the so-called weak asymptotic homomorphism property, which is equivalent to $L(F_d)$ being a weakly mixing type $\II_1$ subfactor. As a consequence, $L(F_d)$ is a singular type $\II_1$ subfactor in $L(\mathscr{T}_d(G_*))$ (i.e., the normalizer of $L(F_d)$ in $L(\mathscr{T}_d(G_*))$ generates $L(F_d)$) and the inclusion $L(F_d) \subseteq L(\mathscr{T}_d(G_*))$ is irreducible, which allows us to upgrade factoriality to $L(\mathscr{T}_d(G_*))$, i.e., $L(\mathscr{T}_d(G_*))$ will also be a type $\II_1$ factor. With a few other natural assumptions on the $d$-ary cloning system, we also give sufficient conditions for when these Thompson-like groups coming from cloning systems yield type $\II_1$ McDuff factor von Neumann algebras and hence inner amenable groups. In particular we prove the surprising result that the pure braided Higman-Thompson group $bF_d$ is inner amenable. Along the way, we also prove that the Higman-Thompson groups $T_d$ and $V_d$ are non-inner amenable, which is an improvement and extension of Haagerup and Olesen's result that $T$ and $V$ are non-inner amenable. We prove all these and a variety of other results in this thesis.
Recommended Citation
Bashwinger, Eli Mackenzie, "Von Neumann Algebras Of Thompson-Like Groups From Cloning Systems" (2024). Legacy Theses & Dissertations (2009 - 2024). 3288.
https://scholarsarchive.library.albany.edu/legacy-etd/3288
