A discreteness criterion for groups containing parabolic isometries

TL;DR

This paper establishes a geometric criterion for the discreteness of hyperbolic isometry groups with parabolic elements, generalizing classical results and providing improved bounds in higher dimensions.

Contribution

It introduces a new geometric discreteness criterion for hyperbolic groups with parabolic isometries, extending known results to higher dimensions with better asymptotic bounds.

Findings

Provides a geometric proof of discreteness criterion

Generalizes classical results to higher dimensions

Offers improved asymptotic bounds

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.