Limit sets of cyclic quaternionic Kleinian groups

TL;DR

This paper studies the limit sets of cyclic subgroups of quaternionic Kleinian groups acting on quaternionic projective space, computing two types of limit sets introduced by Kulkarni and Conze-Guivarc'h.

Contribution

It provides the first explicit computation of Kulkarni and Conze-Guivarc'h limit sets for cyclic subgroups in quaternionic Kleinian groups.

Findings

Computed Kulkarni limit sets for cyclic subgroups

Computed Conze-Guivarc'h limit sets for cyclic subgroups

Enhanced understanding of quaternionic Kleinian group dynamics

Abstract

In this paper, we consider the natural action of $\mathrm{SL}(3, \mathbb{H})$ on the quaternionic projective space $ \mathbb{P}_{\mathbb{H}}^2$. Under this action, we investigate limit sets for cyclic subgroups of $\mathrm{SL}(3, \mathbb{H})$. We compute two types of limit sets, which were introduced by Kulkarni and Conze-Guivarc'h, respectively.