Isomorphisms of Brin-Higman-Thompson groups

Abstract

Let $m,m^{\prime },r,r^{\prime },t,t^{\prime }$ be positive integers with $r,r^{\prime }\ge 2$. 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 $P{U}_m(L_r^{\otimes t})$ denote the group of positive unitary elements. We show that $P{U}_m(L_r^{\otimes t})$ is isomorphic to the Brin-Higman-Thompson group $t{V}_{r,m}$; the case $t=1$ was found by Pardo, that is, $P{U}_m(L_r)$ is isomorphic to the Higman-Thompson group $V_{r,m}$. We survey arguments of Abrams, Ánh, Bleak, Brin, Higman, Lanoue, Pardo, and Thompson that prove that $t^{\prime }{V}_{r^{\prime },m^{\prime }}\cong t{V}_{r,m}$ if and only if $r^{\prime }=r$, $t^{\prime }=t$ and $\gcd (m^{\prime },r^{\prime }-1)=\gcd (m,r-1)$ (if and only if $M_{m^{\prime }}(L_{r^{\prime }}^{\otimes t^{\prime }})$ and $M_m(L_r^{\otimes t})$ are isomorphic as partially ordered rings with involution).

24 pages

References in corpus (2)

• Higher Dimensional Thompson Groups
• On the Baker's map and the Simplicity of the Higher Dimensional Thompson Groups nV

Cited by in corpus (2)

• A short proof of Rubin's theorem
• Generalizations of Loday's assembly maps for Lawvere's algebraic theories