An important problem in quaternionic hyperbolic geometry is to classify ordered $m$-tuples of pairwise distinct points in the closure of quaternionic hyperbolic n-space, $overline{{bf H}_bh^n}$, up to congruence in the holomorphic isometry group ${rm
PSp}(n,1)$ of ${bf H}_bh^n$. In this paper we concentrate on two cases: $m=3$ in $overline{{bf H}_bh^n}$ and $m=4$ on $partial{bf H}_bh^n$ for $ngeq 2$. New geometric invariants and several distance formulas in quaternionic hyperbolic geometry are introduced and studied for this problem. The congruence classes are completely described by quaternionic Cartans angular invariants and the distances between some geometric objects for the first case. The moduli space is constructed for the second case.
In the paper (Osaka J. Math. {bf 46}: 403-409, 2009), Yang conjectured that a non-elementary subgroup $G$ of $SL(2, bc)$ containing elliptic elements is discrete if for each elliptic element $gin G$ the group $< f, g >$ is discrete, where $fin SL(2,b
c)$ is a test map which is loxodromic or elliptic. The purpose of this paper is to give an affirmative answer to this question.
In this paper, we obtain analogues of Jorgensens 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}rgensens 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].
In this paper, we give an analogue of Jorgensens inequality for non-elementary groups of isometries of quaternionic hyperbolic space generated by two elements, one of which is elliptic. As an application, we obtain an analogue of Jorgensens inequality in 2-dimensional Mobius group of the above case.
Jorgensens 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 Jorgensens inequality for non-elementary groups of isometries of quaternionic hyp
erbolic n-space generated by two elements, one of which is loxodromic.
