Extremality of Quaternionic J{\o}rgensen Inequality

Abstract

Let ${\rm SL(2, \mathbb H)}$ be the group of $2 \times 2$ quaternionic matrices with Dieudonn\'e determinant $1$. The group ${\rm SL(2, \mathbb H)}$ acts on the five dimensional hyperbolic space by isometries. We investigate extremality of J\o{}rgensen type inequalities in ${\rm SL(2, \mathbb H)}$. Along the way, we derive J\o{}rgensen type inequalities for quaternionic M\"obius transformations which extend earlier inequalities obtained by Waterman and Kellerhals.