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arXiv · 2604.00466

Convex cocompact groups with three-dimensional limit sets

Sami Douba·Gye-Seon Lee·Ludovic Marquis·Lorenzo Ruffoni
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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.

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Sami Douba, Gye-Seon Lee, Ludovic Marquis, Lorenzo Ruffoni. 2026-04-01. Convex cocompact groups with three-dimensional limit sets. https://arxiv.org/abs/2604.00466

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