Isomorphisms of Brin-Higman-Thompson groups

Abstract

Let m, m′, r, r′, t, t′ be positive integers with r, r′ ⩾ 2. Let $\mathbb{L}_r$ denote the ring that is universal with an invertible 1×r matrix. Let $M_m (\mathbb{L}_r^{ \otimes t} )$ denote the ring of m × m matrices over the tensor product of t copies of $\mathbb{L}_r$. In a natural way, $M_m (\mathbb{L}_r^{ \otimes t} )$ is a partially ordered ring with involution. Let $PU_m (\mathbb{L}_r^{ \otimes t} )$ denote the group of positive unitary elements. We show that $PU_m (\mathbb{L}_r^{ \otimes t} )$ is isomorphic to the Brin-Higman-Thompson group tV _r,m ; the case t=1 was found by Pardo, that is, $PU_m (\mathbb{L}_r )$ 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′V _r′,m′ ≌ tV _r,m if and only if r′ =r, t′ =t and gcd(m′, r′−1) = gcd(m, r−1) (if and only if $M_{m'} (\mathbb{L}_{r'}^{ \otimes t'} )$ and $M_m (\mathbb{L}_r^{ \otimes t} )$ are isomorphic as partially ordered rings with involution).