Extremality of Quaternionic J{\o}rgensen Inequality

TL;DR

This paper explores the extremality conditions of Jørgensen type inequalities within the quaternionic special linear group, extending previous inequalities to quaternionic Möbius transformations acting on hyperbolic space.

Contribution

It introduces new Jørgensen type inequalities for quaternionic Möbius transformations, extending earlier results by Waterman and Kellerhals, and investigates their extremality in SL(2, H).

Findings

Derived new inequalities extending previous results.

Analyzed extremality conditions for quaternionic transformations.

Connected inequalities to hyperbolic geometry actions.

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.