Jorgensen's Inequalities and Collars in n-dimensional Quaternionic Hyperbolic Space

TL;DR

This paper extends Jorgensen's inequality to quaternionic hyperbolic spaces, providing new bounds for isometry groups and applications to geometric structures like collars around geodesics.

Contribution

It develops quaternionic analogues of Jorgensen's inequality, improving previous results and applying them to construct disjoint collars in quaternionic hyperbolic manifolds.

Findings

Derived new inequalities for quaternionic hyperbolic isometry groups.

Constructed disjoint collars around short geodesics in quaternionic hyperbolic manifolds.

Improved bounds over previous results and answered an open question.

Abstract

In this paper, we obtain 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. Our result gives some improvement over earlier results of Kim [10] and Markham [15]}. These results also apply to complex hyperbolic space and give improvements on results of Jiang, Kamiya and Parker [7] As applications, we use the quaternionic version of J{\o}rgensen's inequalities to construct embedded collars about short, simple, closed geodesics in quaternionic hyperbolic manifolds. We show that these canonical collars are disjoint from each other. Our results give some improvement over earlier results of Markham and Parker and answer an open question posed in [16].