REGULAR GRAPHS AND DISCRETE SUBGROUPS OF PROJECTIVE LINEAR GROUPS

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 Γ of the projective linear group such that T /Γ is isomorphic to the graph.