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On signatures of the atoroidal bundles of Kent-Leininger

Jean-François Lafont, Nicholas Miller, Lorenzo Ruffoni

Abstract

We show that there are infinitely many homeomorphism types of atoroidal surface bundles over surfaces which have signature zero.

On signatures of the atoroidal bundles of Kent-Leininger

Abstract

We show that there are infinitely many homeomorphism types of atoroidal surface bundles over surfaces which have signature zero.

Paper Structure

This paper contains 12 sections, 18 theorems, 28 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Hyperbolic manifolds and surface bundles
  4. Recollections on signature
  5. Finding type-preserving lifts of Mod(T2X)
  6. Theorem \ref{['thm:injection_into_closed']} for characteristic covers
  7. Theorem \ref{['thm:injection_into_closed']} for totally irregular covers
  8. The generic case
  9. The once-punctured case
  10. The genus zero case
  11. Proofs of Theorems \ref{['thm:general_irregularcover']} and \ref{['thm:homeotype_2']}
  12. Appendix: bounding null-homologous surfaces

Key Result

Theorem 1.1

For every $g\ge 4$, there exist infinitely many distinct commensurability classes of purely pseudo-Anosov subgroups $\pi_1(S_h)<\mathop{\mathrm{Mod}}\nolimits(S_g)$ where $h\ge 2$. In particular, there are infinitely many homeomorphism types of atoroidal surface bundles over surfaces.

Figures (1)

  • Figure 1: The fundamental group generators from Equation \ref{['eqn:fundgrp']}.

Theorems & Definitions (35)

  • Theorem 1.1: Kent--Leininger
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Theorem 2.2: Earle--Eells
  • Lemma 2.3
  • Theorem 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 25 more