A Decomposition Theorem for Cayley Graphs of Picard Group Quotients

Abstract

The Picard group is defined as $\mathrm{\Gamma }=SL(2,\mathbb{Z}[i])$; the ring of $2\times 2$ matrices with Gaussian integer entries and determinant one. We consider certain graphs associated to quotients $\mathrm{\Gamma }/\mathrm{\Gamma }(p)$ where $p$ is a prime congruent to three mod four and $\mathrm{\Gamma }(p)$ is the congruence subgroup of level $p$. We prove a decomposition theorem on the vertices of these graphs, and use this decomposition to derive upper and lower bounds on their isoperimetric numbers.