Experimental Mathematics | 2021
Quotient Graphs and Amalgam Presentations for Unitary Groups Over Cyclotomic Rings
Abstract
Suppose $4|n$, $n\\geq 8$, $F=F_n=\\mathbb{Q}(\\zeta_n+\\bar{\\zeta}_n)$, and there is one prime $\\mathfrak{p}=\\mathfrak{p}_n$ above $2$ in $F_n$. We study amalgam presentations for $\\operatorname{PU_{2}}(\\mathbb{Z}[\\zeta_n, 1/2])$ and $\\operatorname{PSU_{2}}(\\mathbb{Z}[\\zeta_n, 1/2])$ with the Clifford-cyclotomic group in quantum computing as a subgroup. These amalgams arise from an action of these groups on the Bruhat-Tits tree $\\Delta =\\Delta_{\\mathfrak{p}}$ for $\\operatorname{SL_{2}}(F_\\mathfrak{p})$ constructed via the Hamilton quaternions. We explicitly compute the finite quotient graphs and the resulting amalgams for $8\\leq n\\leq 48$, $n\\neq 44$, as well as for $\\operatorname{PU_{2}}(\\mathbb{Z}[\\zeta_{60}, 1/2])$.