Isomorphisms of Brin-Higman-Thompson groups

© 2014, Hebrew University Magnes Press. 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 (Formula presented.) denote the ring of m × m matrices over the tensor product of t copies of (Formula presented.). In a natural way, (Formula presented.) is a partially ordered ring with involution. Let (Formula presented.) denote the group of positive unitary elements. We show that (Formula presented.) is isomorphic to the Brin-Higman-Thompson group tVr,m; the case t=1 was found by Pardo, that is, $PU_m (\mathbb{L}_r )$ is isomorphic to the Higman-Thompson group Vr,m.We survey arguments of Abrams, Ánh, Bleak, Brin, Higman, Lanoue, Pardo and Thompson that prove that t′Vr′,m′ ≌ tVr,m if and only if r′ =r, t′ =t and gcd(m′, r′−1) = gcd(m, r−1) (if and only if (Formula presented.) are isomorphic as partially ordered rings with involution).