Regular graphs and discrete subgroups of projective linear groups

Abstract

The homothety classes of lattices in a two dimensional vector space over a nonarchimedean local field form a regular tree $\T$ of degree $q+1$ on which the projective linear group acts naturally where $q$ is the order of the residue field. We show that for any finite regular combinatorial graph of even degree $q+1$, there exists a torsion free discrete subgroup $\Gamma$ of the projective linear group such that $\T/\Gamma$ is isomorphic to the graph.