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Convex co-compact representations of 3-manifold groups

Mitul Islam, Andrew Zimmer

Abstract

A representation of a finitely generated group into the projective general linear group is called convex co-compact if it has finite kernel and its image acts convex co-compactly on a properly convex domain in real projective space. We prove that the fundamental group of a closed irreducible orientable 3-manifold can admit such a representation only when the manifold is geometric (with Euclidean, Hyperbolic, or Euclidean $\times$ Hyperbolic geometry) or when every component in the geometric decomposition is hyperbolic. In each case, we describe the structure of such examples.

Convex co-compact representations of 3-manifold groups

Abstract

A representation of a finitely generated group into the projective general linear group is called convex co-compact if it has finite kernel and its image acts convex co-compactly on a properly convex domain in real projective space. We prove that the fundamental group of a closed irreducible orientable 3-manifold can admit such a representation only when the manifold is geometric (with Euclidean, Hyperbolic, or Euclidean Hyperbolic geometry) or when every component in the geometric decomposition is hyperbolic. In each case, we describe the structure of such examples.

Paper Structure

This paper contains 42 sections, 65 theorems, 201 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Euclidean and Euclidean $\times$ hyperbolic manifolds
  3. Hyperbolic manifolds
  4. Non-geometric manifolds
  5. The structure of the domain
  6. Equivariant boundary maps
  7. Dynamics
  8. Recent developments
  9. Preliminaries
  10. Notation
  11. Convexity
  12. Open faces and the Hilbert metric
  13. Cones and simplices
  14. Limits of linear maps
  15. Background on relatively hyperbolic metric spaces
  16. ...and 27 more sections

Key Result

Theorem 1.3

(see Section sec:pf_main_thm below) Suppose $M$ is a closed irreducible orientable 3-manifold. If $\rho : \pi_1(M) \rightarrow \mathop{\mathrm{PGL}}\nolimits_d(\mathop{\mathrm{\mathbb{R}}}\nolimits)$ is a convex co-compact representation, then either

Figures (1)

  • Figure 1: Two asymptotic geodesic $\sigma$ and $\ell_0$ limiting to the $C^1$-smooth point $g^+$

Theorems & Definitions (139)

  • Definition 1.1: Danciger--Guéritaud--Kassel DGK2017
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 129 more