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Thurston norms of tunnel number-one manifolds

Natalia Pacheco-Tallaj, Kevin Schreve, Nicholas G. Vlamis

Abstract

The Thurston norm of a 3-manifold measures the complexity of surfaces representing two-dimensional homology classes. We study the possible unit balls of Thurston norms of 3-manifolds $M$ with $b_1(M) = 2$, and whose fundamental groups admit presentations with two generators and one relator. We show that even among this special class, there are 3-manifolds such that the unit ball of the Thurston norm has arbitrarily many faces.

Thurston norms of tunnel number-one manifolds

Abstract

The Thurston norm of a 3-manifold measures the complexity of surfaces representing two-dimensional homology classes. We study the possible unit balls of Thurston norms of 3-manifolds with , and whose fundamental groups admit presentations with two generators and one relator. We show that even among this special class, there are 3-manifolds such that the unit ball of the Thurston norm has arbitrarily many faces.

Paper Structure

This paper contains 5 sections, 5 theorems, 25 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Algorithm for the Thurston norm
  3. Mapping class groups
  4. Main result
  5. Appendix: Baumslag-Solitar fundamental groups

Key Result

Theorem 1.1

Let $N$ be a $3$-manifold. There exists a unique centrally symmetric marked polytope $\mathcal{M}_N$ in $H_1(N;\mathbb{R})$ such that for any $\phi\in H^1(N;\mathbb{R}) = \mathop{\mathrm{Hom}}\nolimits(H_1(N,\mathbb{R}),\mathbb{R})$ we have Moreover, $\phi$ is fibered if and only if $\phi$ restricted to $\mathcal{M}_N$ attains its maximum on a marked vertex.

Figures (6)

  • Figure 1: The algorithm applied to the group: $\langle x,y \,|\, r=xyxyyxyXYXYYXY\rangle$, where capital letters denote inverses. This is the first case of our sequence of examples in Section \ref{['s:construction']}.
  • Figure 2: The right Dehn twists about the curves $a,b,c,d$, and $e$ generate $\mathop{\mathrm{Mod}}\nolimits(\Sigma_2)$. The loops $w,x,y,$ and $z$ are standard generators for $\pi_1(\Sigma_2)$, which we will use throughout the article. The curve $\gamma$ in $\pi_1(\Sigma_2)$ is the commutator $[x,z]$.
  • Figure 3: A general picture of $g_n([x,z])$, after removing the letters $z$ and $w$. The relator starts at the origin and climbs up the bottom "staircase" which is $g_n(xz)$. Then $g_n(x^{-1}z^{-1})$ travels back down.
  • Figure 4: The marked polytope $\mathcal{M}_{M_{g_n}}$ with $4n$ vertices.
  • Figure 5: Shown here is the manifold $M_{f_m}\cong N = M_1 \cup M_2$ with $m = 4$. The manifold $M_1$ is drawn on the left and $M _2$ on the right. Curves of the same color are identified in $M_{f_m}$.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Theorem 1.1: Thurston
  • Definition 3.1
  • Lemma 3.2
  • Lemma 4.1
  • proof
  • Definition 4.2
  • Theorem 4.3
  • Remark
  • proof : Proof of Theorem \ref{['lem:evenk']}
  • Theorem 4.4
  • ...and 2 more