Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Homomorphisms to 3-manifold groups

Daniel Groves, Michael Hull, Hao Liang

Abstract

We prove foundational results about the set of homomorphisms from a finitely generated group to the collection of all fundamental groups of compact 3-manifolds and answer questions of Reid-Wang-Zhou and Agol-Liu.

Homomorphisms to 3-manifold groups

Abstract

We prove foundational results about the set of homomorphisms from a finitely generated group to the collection of all fundamental groups of compact 3-manifolds and answer questions of Reid-Wang-Zhou and Agol-Liu.

Paper Structure

This paper contains 31 sections, 102 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Outline of proofs
  3. Proof of Theorem \ref{['t:DCC']}
  4. Notation and conventions
  5. Reduction to $\mathcal{M}_{\operatorname{Gen}}^\pi$
  6. Equationally noetherian families
  7. Reduction to closed, orientable, irreducible $3$--manifolds
  8. The geometric decomposition
  9. Reduction to $\mathcal{M}_{\operatorname{Gen}}^\pi$
  10. Generalized geometric decompositions
  11. The geometric decomposition
  12. Assumptions
  13. Outline of the proof of Theorem \ref{['t:collapsing Divergent']}
  14. The Collapsed Space
  15. The collapsed space for Kleinian and Fuchsian groups
  16. ...and 16 more sections

Key Result

Theorem 1

For any finitely generated group $G$ there is a finitely presented group $G_0$ and an epimorphism $\alpha \colon G_0 \twoheadrightarrow G$ so that for every $\Gamma \in \mathcal{M}^\pi$ the map induced by precomposition with $\alpha$ is a bijection.

Theorems & Definitions (207)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • ...and 197 more