Cusps of lattices in rank 1 Lie groups over local fields

Abstract

Let G be the group of rational points of a semisimple algebraic group of rank 1 over a nonarchimedean local field. We improve upon Lubotzky's analysis of graphs of groups describing the action of lattices in G on its Bruhat-Tits tree assuming a condition on unipotents in G. The condition holds for all but a few types of rank 1 groups. A fairly straightforward simplification of Lubotzky's definition of a cusp of a lattice is the key step to our results. We take the opportunity to reprove Lubotzky's part in the analysis from this foundation.