Convex cocompact groups with three-dimensional limit sets

TL;DR

The paper introduces a method to construct convex cocompact hyperbolic reflection groups with three-dimensional limit sets based on arbitrary simplicial complexes, expanding understanding of hyperbolic group actions.

Contribution

It provides a general construction for convex cocompact hyperbolic reflection groups with specified limit sets, answering a question about their existence beyond standard spheres.

Findings

Constructed groups have limit sets as ech cohomology spheres.

The groups are thin subgroups of cocompact arithmetic hyperbolic lattices.

The construction applies to any 3-dimensional simplicial complex.

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.