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Pseudo-Anosov subgroups of surface bundles over tori

Junmo Ryang

Abstract

We show that finitely generated, purely pseudo-Anosov subgroups of the fundamental groups of surface bundles over tori are convex cocompact as subgroups of the mapping class group via the Birman exact sequence. This generalizes the fact that similar groups within fibered 3-manifold groups are convex cocompact, which is a combination of results due to Dowdall, Kent, Leininger, Russell, and Schleimer.

Pseudo-Anosov subgroups of surface bundles over tori

Abstract

We show that finitely generated, purely pseudo-Anosov subgroups of the fundamental groups of surface bundles over tori are convex cocompact as subgroups of the mapping class group via the Birman exact sequence. This generalizes the fact that similar groups within fibered 3-manifold groups are convex cocompact, which is a combination of results due to Dowdall, Kent, Leininger, Russell, and Schleimer.
Paper Structure (32 sections, 34 theorems, 65 equations)

This paper contains 32 sections, 34 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Surface group extensions
  3. Proof Summary
  4. Bounding $T^u \cap T^G$
  5. Bounding $p_0(T^u \cap T^G)$
  6. Acknowledgments
  7. Preliminaries
  8. Curve complexes and arc complexes
  9. Mapping class group and the Birman exact sequence
  10. Fibers and trees
  11. Convex cocompactness
  12. Setup
  13. Realizing $H$ by homeomorphisms
  14. $H$ acting on subsurfaces and annuli
  15. $\Gamma_H$ acting on $\mathbb{H}^2$ and T
  16. ...and 17 more sections

Key Result

Theorem 1.1

Suppose $\chi(S) < 0$ and $E$ is an $S$-bundle over $T^n$. A subgroup $G < \Gamma = \mu^z(\pi_1E)$ is convex cocompact if and only if it is finitely generated and purely pseudo-Anosov.

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.3
  • Lemma 1.4
  • proof
  • Corollary 1.5
  • proof
  • Proposition 2.1: MR2851869MR4632569
  • Theorem 2.2: MR0611385
  • Theorem 2.3: MR2599078
  • Theorem 2.4: MR2465691MR2349677
  • ...and 52 more