A Decomposition Theorem for Cayley Graphs of Picard Group Quotients

The Picard group is defined as \( \Gamma = SL(2, \mathbb{Z}[i]) \); the ring of \( 2 \times 2 \) matrices with Gaussian integer entries and determinant one. We consider certain graphs associated to quotients \( \Gamma/\Gamma(p) \) where \( p \) is a prime congruent to three mod four and \( \Gamma(p) \) 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.