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

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

TL;DR

This paper realizes the Pontryagin sphere as the limit set of convex cocompact subgroups of $\mathrm{Isom}(\mathbb{H}^d)$ by constructing two explicit examples: a 50-reflection subgroup in $\mathrm{Isom}(\mathbb{H}^4)$ with limit set homeomorphic to the Pontryagin sphere, and a 21-generator–derived reflection group in $\mathrm{Isom}(\mathbb{H}^6)$ with the same limit set (containing a finite-index right-angled subgroup). It also identifies convex cocompact subgroups whose limit sets are Menger curves in both dimensions. The constructions use flag-no-square nerves from the $600$-cell and from a torus, together with the Poincaré polyhedron theorem and RACG technology to guarantee convex cocompactness. The results extend Bourdon's earlier Menger-curve examples, discuss arithmeticity and Zariski-density, and place the Pontryagin sphere and Menger-curve limit sets in a unified hyperbolic-group framework.

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.

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

TL;DR

This paper realizes the Pontryagin sphere as the limit set of convex cocompact subgroups of by constructing two explicit examples: a 50-reflection subgroup in with limit set homeomorphic to the Pontryagin sphere, and a 21-generator–derived reflection group in with the same limit set (containing a finite-index right-angled subgroup). It also identifies convex cocompact subgroups whose limit sets are Menger curves in both dimensions. The constructions use flag-no-square nerves from the -cell and from a torus, together with the Poincaré polyhedron theorem and RACG technology to guarantee convex cocompactness. The results extend Bourdon's earlier Menger-curve examples, discuss arithmeticity and Zariski-density, and place the Pontryagin sphere and Menger-curve limit sets in a unified hyperbolic-group framework.

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 reflections of , and the other by a rotation of order and a reflection of . For each of them, we also locate convex cocompact subgroups with limit set a Menger curve.

Paper Structure

This paper contains 15 sections, 12 theorems, 8 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simplicial complexes
  4. The Pontryagin sphere
  5. Right-angled Coxeter groups
  6. Convex cocompactness
  7. Construction in H4
  8. The 600-cell as a union of two solid tori
  9. Proof of Theorem \ref{['main4']}
  10. Construction in H6
  11. An abstract Coxeter group with 21 generators
  12. Construction of the reflection group Gamma6 of Theorem \ref{['main6']}
  13. A two generator group that contains Gamma6 as a finite-index subgroup
  14. A few remarks
  15. Special subgroups with limit set a Menger curve

Key Result

Theorem 1.1

There is a convex cocompact right-angled reflection subgroup $\Gamma^4$ of $\mathrm{Isom} (\mathbb H^4)$ whose limit set is homeomorphic to the Pontryagin sphere.

Figures (6)

  • Figure 1: A rendering of the Pontryagin sphere, courtesy of Theodore Weisman. This is in fact a plot of the limit set of the group $\Gamma^4$ in Theorem \ref{['main4']}.
  • Figure 2: From left to right: a flying saucer, two flying saucers stacked vertex-to-vertex with ten interstitial tetrahedra (indicated in red) around their vertex of intersection, and a drum.
  • Figure 3: The triangulation of $\partial T_{0}$ with 50 vertices, 150 edges, and 100 triangles. Opposite edges are identified. This defines a RACG generated by 50 reflections inside $W_{\mathsf{C}_{600}}$.
  • Figure 4: The nerve of the Coxeter group $W_{T_{21}}$
  • Figure 5: A special subgroup of $\Gamma^d$ with limit set a Menger curve ($d=4$ left, $d=6$ right).
  • ...and 1 more figures

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.2
  • ...and 17 more