On discreteness of subgroups of quaternionic hyperbolic isometries

Abstract

Let HnH denote the n -dimensional quaternionic hyperbolic space. The linear group Sp(n,1) acts on HnH by isometries. A subgroup G of Sp(n,1) is called Zariski dense if it neither fixes a point on HnH∪∂HnH nor preserves a totally geodesic subspace of HnH . We prove that a Zariski dense subgroup G of Sp(n,1) is discrete if for every loxodromic element g∈G the two-generator subgroup ⟨f,gfg−1⟩ is discrete, where the generator f∈Sp(n,1) is a certain fixed element not necessarily from G .