Commensurability invariants for nonuniform tree lattices

Abstract

We study nonuniform lattices in the automorphism group G of a locally finite simplicial tree X. In particular, we are interested in classifying lattices up to commensurability in G. We introduce two new commensurability invariants: quotient growth, which measures the growth of the noncompact quotient of the lattice; and stabilizer growth, which measures the growth of the orders of finite stabilizers in a fundamental domain as a function of distance from a fixed basepoint. When X is the biregular tree X_{m,n}, we construct lattices realizing all triples of covolume, quotient growth, and stabilizer growth satisfying some mild conditions. In particular, for each positive real number \nu we construct uncountably many noncommensurable lattices with covolume \nu.