On discreteness of subgroups of quaternionic hyperbolic isometries

Let ${{\bf H}_{\mathbb H}}^n$ denote the $n$-dimensional quaternionic hyperbolic space. The linear group ${\rm{Sp}}(n,1)$ acts by the isometries of ${{\bf H}_{\mathbb H}}^n$. A subgroup $G$ of ${\rm {Sp}}(n,1)$ is called \emph{Zariski dense} if it does not fix a point on ${{\bf H}_{\mathbb H}}^n \cup \partial {{\bf H}_{\mathbb H}}^n$ and neither it preserves a totally geodesic subspace of ${{{\bf H}}_{\mathbb H}}^n$. We prove that a Zariski dense subgroup $G$ of ${\rm{ Sp}}(n,1)$ is discrete if for every loxodromic element $g \in G$ the two generator subgroup $\langle f, g f g^{-1} \rangle$ is discrete, where the generator $f \in {\rm{Sp}}(n,1)$ is certain fixed element not necessarily from $G$.Comment: Reformatted, adding new result, and removing some. Removed parts will be subsumed elsewher

Green’s Function for a Slice of the Korányi Ball in the Heisenberg Group H

We give a representation formula for solution of the inhomogeneous Dirichlet problem on the upper half Korányi ball and for the slice of the Korányi ball in the Heisenberg group Hn by obtaining explicit expressions of Green-like kernel when the given data has certain radial symmetry