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Convex cocompact groups with three-dimensional limit sets

Sami Douba, Gye-Seon Lee, Ludovic Marquis, Lorenzo Ruffoni

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-Świątkowski 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 Čech cohomology sphere other than the standard sphere.

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-Świątkowski 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 Čech cohomology sphere other than the standard sphere.

Paper Structure

This paper contains 18 sections, 8 theorems, 14 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simplicial complexes and the subdivisions of Dranishnikov and Przytycki--Świątkowski
  4. Right-angled Coxeter groups
  5. Some low-dimensional continua
  6. Trees of manifolds
  7. Pontryagin surfaces
  8. Quasiconvex subgroups with limit set a Menger curve
  9. Proof of Theorem \ref{['thm:main']}
  10. A particularly nice Lorentzian bilinear form
  11. Passing to a larger complex
  12. Generating the reflection group $\Gamma_n$
  13. The nerve of $\Gamma_n$ is $L_n^{\#}$
  14. Pairs of vertices from a single $3$-simplex of $L_n$
  15. Pairs of vertices from distinct $3$-simplices of $L_n$
  16. ...and 3 more sections

Key Result

Theorem 1.2

For each $n \geqslant 4$, there is a cocompact arithmetic lattice $\Delta_n < \mathrm{Isom}(\mathbb H^n)$ such that, for any simplicial complex $L$ on $n$ vertices and of dimension $d \leqslant 3$, the lattice $\Delta_n$ contains a right-angled reflection subgroup $\Gamma_L$ whose nerve is $L^\#$. M

Figures (3)

  • Figure 1: Dranishnikov's subdivision procedure for 2-dimensional simplicial complexes.
  • Figure 2: (Left) A $2$-simplex $\sigma$ of $L_n$; (Right) the subdivision $\sigma^\#$ of $\sigma$ in $L_n^\#$.
  • Figure 3: Triangulations of $\widehat{M}_2$, $\widehat{M}_3$, and $\widehat{M}_5$, shown from left to right.

Theorems & Definitions (16)

  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Remark 1.6
  • Corollary 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2: Theorem 3.7 in FI03, Theorem 2 in SW20
  • Lemma 2.3: Proof of DR97
  • ...and 6 more