Isomorphisms of Brin-Higman-Thompson groups

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).