Convex cocompact groups in real hyperbolic spaces with limit set a Pontryagin sphere

TL;DR

The paper presents two examples of convex cocompact subgroups in real hyperbolic spaces with limit sets being Pontryagin spheres, expanding understanding of such groups' geometric structures.

Contribution

It provides explicit constructions of convex cocompact groups with Pontryagin sphere limit sets in higher-dimensional hyperbolic spaces.

Findings

Constructed a group generated by 50 reflections in H^4 with Pontryagin sphere limit set.

Constructed a group with a rotation of order 21 and a reflection in H^6 with Pontryagin sphere limit set.

Located convex cocompact subgroups with Menger curve limit sets for each example.

Abstract

We exhibit two examples of convex cocompact subgroups of the isometry groups of real hyperbolic spaces with limit set a Pontryagin sphere: one generated by $50$ reflections of $\mathbb{H}^4$, and the other by a rotation of order $21$ and a reflection of $\mathbb{H}^6$. For each of them, we also locate convex cocompact subgroups with limit set a Menger curve.