Warren Dicks and Conchita Martínez-Pérez (2011)

# Isomorphisms of Brin-Higman-Thompson groups

arXiv:1112.1606.

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\_rˆ\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\_rˆ\otimes t\)\$ is a partially ordered ring with involution. Let \$PU\_m(L\_rˆ\otimes t\)\$ denote the group of positive unitary elements. We show that \$PU\_m(L\_rˆ\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'\(L\_\r'\ˆ\otimes t'\)\$ and \$M\_m(L\_rˆ\otimes t\)\$ are isomorphic as partially ordered rings with involution).

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