Isomorphisms of Brin-Higman-Thompson groups

TL;DR

This paper establishes isomorphisms between certain Brin-Higman-Thompson groups and groups of positive unitaries over tensor products of rings, extending known results and classifying these groups via ring isomorphisms.

Contribution

It generalizes the isomorphism between Higman-Thompson groups and positive unitary groups to a broader class involving tensor products of rings, providing a classification criterion.

Findings

$PU_m(L_r^{ ensor t})$ is isomorphic to $t V_{r,m}$

Classification of groups based on ring isomorphisms and gcd conditions

Extension of Pardo's case to tensor product rings

Abstract

Let $m, m', r, r',t, t'$ be positive integers with $r, r' \ge 2$. Let $L_r$ denote the ring that is universal with an invertible $1 \times r$ matrix. Let $M_m(L_r^{\otimes t})$ denote the ring of $m \times m$ matrices over the tensor product of $t$ copies of $L_r$. In a natural way, $M_m(L_r^{\otimes t})$ is a partially ordered ring with involution. Let $PU_m(L_r^{\otimes t})$ denote the group of positive unitary elements. We show that $PU_m(L_r^{\otimes t})$ is isomorphic to the Brin-Higman-Thompson group $t V_{r,m}$; the case $t =1$ was found by Pardo, that is, $PU_m(L_r)$ is isomorphic to the Higman-Thompson group $V_{r,m}$. We survey arguments of Abrams, \'Anh, Bleak, Brin, Higman, Lanoue, Pardo, and Thompson that prove that $t' V_{r',m'} \cong tV_{r,m} $ if and only if $r' = r$, $t'=t$ and $ \gcd(m',r'-1) = \gcd(m,r-1)$ (if and only if $M_{m'}(L_{r'}^{\otimes t'})$ and $M_m(L_r^{\otimes t})$ are isomorphic as partially ordered rings with involution).