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Detecting embedded surfaces using finite quotients

Tam Cheetham-West, Khanh Le

Abstract

We give conditions on a Haken hyperbolic rational homology three sphere that imply that any other 3-manifold with profinitely equivalent fundamental group must also be Haken. In the appendix, we show that a regular finite-sheeted cover of an aspherical integral homology three-sphere with positive first Betti number must have first Betti number at least four. We also show that this lower bound is sharp.

Detecting embedded surfaces using finite quotients

Abstract

We give conditions on a Haken hyperbolic rational homology three sphere that imply that any other 3-manifold with profinitely equivalent fundamental group must also be Haken. In the appendix, we show that a regular finite-sheeted cover of an aspherical integral homology three-sphere with positive first Betti number must have first Betti number at least four. We also show that this lower bound is sharp.
Paper Structure (20 sections, 28 theorems, 6 equations, 1 figure)

This paper contains 20 sections, 28 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on profinite completions
  3. Generalities
  4. The case of 3-manifold groups
  5. Reducing to the case of hyperbolic ${\mathbb Q}$H$S^3$s
  6. Proof of the main theorem
  7. The relationship between conditions in \ref{['thm:mainhakentheorem']} and Haken
  8. When $M$ has an embedded virtual fiber or $\pi_1(M)$ has an infinite dihedral quotient
  9. Property $\mathcal{H}$
  10. When $M$ has a large $\mathop{\mathrm{SL}}\nolimits_2$-character variety over $\mathbb{C}$
  11. When $M$ has a large $\mathop{\mathrm{SL}}\nolimits_2$-character variety over the field with positive characteristic.
  12. The case of non-integral trace
  13. Closed hyperbolic $\mathbb{Q}HS^3$s with Haken profinite genus
  14. Calegari--Dunfield tower of hyperbolic $\mathbb{Q}HS^3$
  15. ${\mathbb Q} HS^3$ with large $\mathop{\mathrm{SL}}\nolimits_2$-character variety over ${\mathbb C}$
  16. ...and 5 more sections

Key Result

Theorem 1.3

Let $M$ be a Haken, hyperbolic ${\mathbb Q}$H$S^3$ and let $N$ be a 3-manifold with $\widehat{\pi_1(M)}\cong\widehat{\pi_1(N)}$. If one of the following holds: then $N$ is Haken.

Figures (1)

  • Figure 1: The diagram summarizes the implication among the conditions in \ref{['thm:mainhakentheorem']} as well as the property of being Haken of a 3-manifold. Here, the (4) condition is referred to as having a large character variety. This terminology comes from the fact that a knot exterior is large if it contains a closed embedded essential surface. Finally, "$\exists$ ANI character" denotes the last condition of \ref{['thm:mainhakentheorem']} in which "ANI" strands for algebraic non-integral.

Theorems & Definitions (40)

  • Definition 1.2
  • Theorem 1.3
  • Proposition 2.1: DFPR,RibesZalesskiiBook
  • Lemma 2.2
  • Corollary 2.3
  • Definition 2.4
  • Theorem 2.5: WZ2
  • Theorem 2.6: WZ2
  • Theorem 2.7: LiuAlmostProfiniteRigidity
  • Lemma 3.1
  • ...and 30 more