Jørgensen’s Inequalities and Collars in n-dimensional Quaternionic Hyperbolic Space

Abstract: In this paper, we obtain analogues of Jørgensen’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 [9], Markham [14] and Cao [2]. 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 Jørgensen’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 [15].