A discreteness criterion for groups containing parabolic isometries

Abstract

This note will prove a discreteness criterion for groups of orientation-preserving isometries of the hyperbolic space which contain a parabolic element. It can be viewed as a generalization of the well-known results of Shimizu-Leutbecher and Jorgensen in dimensions 2 and 3, and is closely related to Waterman's inequality in higher dimensions. Unlike his algebraic method, the argument presented here is geometric and yields an improved asymptotic bound.

Substantially revised version, now covering all dimensions. 8 pages, 1 figure