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

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

2009-06-19

arxiv.org/abs/0906.3562v4

Mathematics

Geometric Topology

19 pages,1 figures, To appear in The Quarterly Journal Of Mathematics

Scientific paper

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].

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