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

Abstract

Let SL(2, $\mathbb{H}$) be the group of $2\times 2$ quaternionic matrices $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ with quaternionic determinant $\det A=|ad-ac{a}^{-1}b|=1$. 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.

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