Discreteness Of Hyperbolic Isometries by Test Maps

Abstract

Let F=R, C or H. Let HnF denote the n-dimensional F-hyperbolic space. Let U(n,1;F) be the linear group that acts by the isometries. A subgroup G of U(n,1;F) is called \emph{Zariski dense} if it does not fix a point on the closure of the F-hyperbolic space, and neither it preserves a totally geodesic subspace of it. We prove that a Zariski dense subgroup G of U(n,1;F) is discrete if for every loxodromic element g∈G, the two generator subgroup ⟨f,g⟩ is discrete, where f∈U(n,1;F) is a test map not necessarily from G.