Convex cocompact groups with three-dimensional limit sets
Abstract
We provide a general construction of convex cocompact hyperbolic reflection groups with three-dimensional limit sets. More precisely, our construction takes as input an arbitrary simplicial complex L of dimension 3 on n vertices, and outputs a convex cocompact right-angled reflection group acting on real hyperbolic n-space whose nerve is precisely the Przytycki-\'Swi\k{a}tkowski subdivision of L. Moreover, the output reflection group is a thin subgroup of an n-dimensional cocompact arithmetic hyperbolic lattice. This answers affirmatively a question of M. Kapovich concerning the existence of a convex cocompact group acting on some real hyperbolic space with limit set a \v{C}ech cohomology sphere other than the standard sphere.
Cite
@article{arxiv.2604.00466,
title = {Convex cocompact groups with three-dimensional limit sets},
author = {Sami Douba and Gye-Seon Lee and Ludovic Marquis and Lorenzo Ruffoni},
journal= {arXiv preprint arXiv:2604.00466},
year = {2026}
}
Comments
14 pages, 3 figures