Quotient graphs and amalgam presentations for unitary groups over cyclotomic rings

Abstract

Suppose $4|n$, $n\ge 8$, $F=F_n=\mathbb{Q}({\zeta }_n+{\bar{\zeta }}_n)$, and there is one prime $\mathfrak{p}={\mathfrak{p}}_n$ above $2$ in $F_n$. We study amalgam presentations for $\mathrm{P}{\mathrm{U}}_2(\mathbb{Z}[{\zeta }_n,1/2])$ and $\mathrm{PS}{\mathrm{U}}_2(\mathbb{Z}[{\zeta }_n,1/2])$ 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 $\mathrm{S}{\mathrm{L}}_2(F_{\mathfrak{p}})$ constructed via the Hamilton quaternions. We explicitly compute the finite quotient graphs and the resulting amalgams for $8\le n\le 48$, $n\ne 44$, as well as for $\mathrm{P}{\mathrm{U}}_2(\mathbb{Z}[{\zeta }_{60},1/2])$.