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Strongly Dense Representations of Hyperbolic 3-Manifold Groups

Ricky Lee

Abstract

We provide the first examples of strongly dense representations of a hyperbolic 3-manifold group into $SL(4,\mathbb{R})$ and $SU(3,1)$ i.e. representations where every pair of non-commuting elements has Zariski dense image. Our examples are holonomy representations arising from projective deformations of its hyperbolic structure. As a Corollary, we get that $SL(4,\mathbb{R})$ has non-Hitchin strongly dense surface subgroups.

Strongly Dense Representations of Hyperbolic 3-Manifold Groups

Abstract

We provide the first examples of strongly dense representations of a hyperbolic 3-manifold group into and i.e. representations where every pair of non-commuting elements has Zariski dense image. Our examples are holonomy representations arising from projective deformations of its hyperbolic structure. As a Corollary, we get that has non-Hitchin strongly dense surface subgroups.
Paper Structure (4 sections, 13 theorems, 31 equations)

This paper contains 4 sections, 13 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Irreducibility
  3. Complex Hyperbolic Geometry
  4. Proof of Theorems

Key Result

Theorem 1.1

The representation $\rho_v:\Gamma_3\rightarrow SL(4,\mathbb{R})$ is strongly dense for all but a possibly countable subset of $v\in (2,\infty)$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Claim 1
  • ...and 21 more