Jorgensen's inequality for quaternionic hyperbolic n-space

TL;DR

This paper extends Jorgensen's inequality, originally for real hyperbolic space, to quaternionic hyperbolic n-space, providing necessary conditions for the discreteness of two-generator isometry groups involving a loxodromic element.

Contribution

It introduces analogues of Jorgensen's inequality for quaternionic hyperbolic spaces, broadening the understanding of group discreteness criteria in higher-dimensional hyperbolic geometry.

Findings

Derived inequalities for quaternionic hyperbolic isometry groups

Established conditions for non-elementary groups generated by two elements

Extended classical results to quaternionic hyperbolic n-space

Abstract

Jorgensen's inequality gives a necessary condition for a non-elementary two generator group of isometries of real hyperbolic 2-space to be discrete. We give analogues of Jorgensen's inequality for non-elementary groups of isometries of quaternionic hyperbolic n-space generated by two elements, one of which is loxodromic.