Test map and Discreteness in SL(2, $\mathbb H$)

Abstract

Let SL(2, $\mathbb H$) be the group of $2 \times 2$ quaternionic matrices $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with quaternionic determinant $\det A=|ad-aca^{-1} b|=1$. This group acts by the orientation-preserving isometries of the five dimensional (real) hyperbolic space. We obtain discreteness criteria for Zariski-dense subgroups of SL(2, $\mathbb H$) using test maps.