Quotient Graphs and Amalgam Presentations for Unitary Groups Over Cyclotomic Rings

Abstract

Suppose $4|n,n\ge 8$, $F=F_n=\mathbf{Q}({\zeta }_n+{\overline{\zeta }}_n)$, and there is one prime $\mathrm{p}={\mathrm{p}}_n$ above 2 in Fn. We study amalgam presentations for ${\text{PU}}_2(\mathbf{Z}[{\zeta }_n,1/2])$ and ${\text{PSU}}_2(\mathbf{Z}[{\zeta }_n,1/2])$ 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 $\Delta ={\Delta }_{\mathrm{p}}$ for ${\text{SL}}_2(F_{\mathrm{p}})$ 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 ${\text{PU}}_2(\mathbf{Z}[{\zeta }_{60},1/2])$.

Keywords:

• unitary groups
• cyclotomic rings
• quotient graphs
• trees
• Clifford-cyclotomic group
• amalgams
• quantum computing

2010 Mathematics Subject Classification.:

• Primary: 20G30
• Secondary: 11R18
• 81P45