Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convex co-compact groups with one dimensional boundary faces

Mitul Islam, Andrew Zimmer

Abstract

In this paper we consider convex co-compact subgroups of the projective linear group. We prove that such a group is relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank two if and only if each open face in the ideal boundary has dimension at most one. We also introduce the "coarse Hilbert dimension" of a subset of a convex set and use it to characterize when a naive convex co-compact subgroup is word hyperbolic or relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank two.

Convex co-compact groups with one dimensional boundary faces

Abstract

In this paper we consider convex co-compact subgroups of the projective linear group. We prove that such a group is relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank two if and only if each open face in the ideal boundary has dimension at most one. We also introduce the "coarse Hilbert dimension" of a subset of a convex set and use it to characterize when a naive convex co-compact subgroup is word hyperbolic or relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank two.

Paper Structure

This paper contains 25 sections, 36 theorems, 150 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Naive convex co-compact subgroups
  3. Preliminaries
  4. Convexity
  5. The Hilbert metric and faces
  6. The center of mass of a compact subset
  7. The Hausdorff distance
  8. Local Hausdorff convergence topology
  9. Properly embedded simplices
  10. Relative hyperbolic convex co-compact groups
  11. General relatively hyperbolic groups
  12. Convex co-compact relatively hyperbolic groups
  13. Properties of coarse dimension
  14. Proof of Theorem \ref{['thm:ncc_GH']}
  15. Proof of Theorem \ref{['thm:cc_one_d_faces']}
  16. ...and 10 more sections

Key Result

Theorem 1.4

Suppose $\Omega \subset \mathop{\mathrm{\mathbb{P}}}\nolimits(\mathop{\mathrm{\mathbb{R}}}\nolimits^d)$ is a properly convex domain, $\Gamma \subset \mathop{\mathrm{Aut}}\nolimits(\Omega)$ is convex co-compact, and $\mathop{\mathrm{\mathcal{C}}}\nolimits : = \mathop{\mathrm{\mathcal{C}}}\nolimits_\O

Figures (1)

  • Figure 1: Figure for the proof of part (4) Case 4, when the convex hull of $\overline{\ell}_1 \cup \overline{\ell}_2$ is a two-dimensional 4-gon.

Theorems & Definitions (87)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4: Danciger--Guéritaud--Kassel DGF2017
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9: Benoist B2006
  • Corollary 1.10: to Benoist's result
  • ...and 77 more