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A Simple Computation of Teichmüller Polynomials from Integer Permutations

Ahmad Rafiqi

TL;DR

The paper presents a streamlined method to compute Teichmüller polynomials for fibered faces of mapping tori of pseudo-Anosov maps with orientable foliations that fix singular trajectories, using Ordered Block Permutations to encode the combinatorics and McMullen's train-track framework. It reduces the computation to a single-vertex zippered-rectangle setup and derives a concrete, implementable formula $\Theta_F(\mathbf{t},u)=\frac{\det(uI-A(\mathbf{t}))}{u-1}$ from a lifted train-track on a Galois cover. The authors construct an infinite family of Teichmüller polynomials $\Theta_{g,p}$ for genus $g\ge2$ and $p\ge0$, obtaining explicit closed-form expressions and demonstrating that these polynomials realize a positive proportion of bi-Perron units as pseudo-Anosov stretch-factors. They also analyze the associated Teichmüller norm and fibered cone, showing the geometric and arithmetic structure of the realized monodromies within the fibered cones. The work highlights the practicality of OBPs for explicit Teichmüller polynomial computations and contributes to understanding the distribution of bi-Perron stretch factors across families of mapping tori.

Abstract

We present a simple method to compute the Teichmüller polynomial of the fibered face of a hyperbolic $3$-manifold $M_φ$ obtained as the mapping torus of a pseudo-Anosov homeomorphism $φ$ of a closed surface. We assume $φ$ has orientable invariant foliations and fixes each singular trajectory. We use a characterisation of such homeomorphisms in terms of a permutation of a finite set of integers to give a direct implementation of McMullens algorithm using train tracks. Train tracks with a single vertex suffice in this case. As an application, for each $p\in\mathbb{Z}_{\geq0}$, we find an infinite sequence of Teichmüller polynomials $Θ_{g,p}$ associated to pseudo-Anosov maps on surfaces of genus $g\geq2$, such that the hyperbolic 3-manifold obtained as the mapping torus has first Betti number $g$. These polynomials realize a positive proportion of bi-Perron units of each degree as pseudo-Anosov stretch-factors.

A Simple Computation of Teichmüller Polynomials from Integer Permutations

TL;DR

The paper presents a streamlined method to compute Teichmüller polynomials for fibered faces of mapping tori of pseudo-Anosov maps with orientable foliations that fix singular trajectories, using Ordered Block Permutations to encode the combinatorics and McMullen's train-track framework. It reduces the computation to a single-vertex zippered-rectangle setup and derives a concrete, implementable formula from a lifted train-track on a Galois cover. The authors construct an infinite family of Teichmüller polynomials for genus and , obtaining explicit closed-form expressions and demonstrating that these polynomials realize a positive proportion of bi-Perron units as pseudo-Anosov stretch-factors. They also analyze the associated Teichmüller norm and fibered cone, showing the geometric and arithmetic structure of the realized monodromies within the fibered cones. The work highlights the practicality of OBPs for explicit Teichmüller polynomial computations and contributes to understanding the distribution of bi-Perron stretch factors across families of mapping tori.

Abstract

We present a simple method to compute the Teichmüller polynomial of the fibered face of a hyperbolic -manifold obtained as the mapping torus of a pseudo-Anosov homeomorphism of a closed surface. We assume has orientable invariant foliations and fixes each singular trajectory. We use a characterisation of such homeomorphisms in terms of a permutation of a finite set of integers to give a direct implementation of McMullens algorithm using train tracks. Train tracks with a single vertex suffice in this case. As an application, for each , we find an infinite sequence of Teichmüller polynomials associated to pseudo-Anosov maps on surfaces of genus , such that the hyperbolic 3-manifold obtained as the mapping torus has first Betti number . These polynomials realize a positive proportion of bi-Perron units of each degree as pseudo-Anosov stretch-factors.
Paper Structure (20 sections, 6 theorems, 64 equations, 3 figures)

This paper contains 20 sections, 6 theorems, 64 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Pseudo-Anosov Maps
  4. Fibered $3$-Manifolds and the Thurston Norm
  5. Teichmüller Polynomials
  6. Zippered Rectangles
  7. Ordered Block Permutations
  8. The Algorithm
  9. The Galois Cover
  10. The Train Track $\widetilde{\tau}$
  11. Example of Fig. \ref{['example']}, part 2/3:
  12. Computing the Teichmüller Polynomial
  13. Proof of Theorem \ref{['mainThm']}
  14. The example of Fig. \ref{['example']}, part 3/3:
  15. Sequences of Teichmüller Polynomials
  16. ...and 5 more sections

Key Result

Proposition 1

Let $m\geq2$ and $a_1,\cdots,a_{m-1}$ be non-negative integers such that is a primitive integer vector. Then, for each $a_m\geq3+2(a_1+\cdots+a_{m-1})$, the largest real root of is a bi-Perron unit and is the stretch-factor of a pseudo-Anosov map on a connected surface.

Figures (3)

  • Figure 1: $x^4+ax^3+bx^2+ax+1=0$ defines a bi-Perron unit iff $(a,b)\in\mathbb{Z}^2$ belongs to the shaded areas shown. Among these, those with $b\leq2a-3$ are shown to be realized as pseudo-Anosov stretch-factors on connected surfaces by Proposition \ref{['Prop1']}.
  • Figure 2: An example, with $n=2g=6$, $\sigma=(2,4,1,6,3,5)$ and $\mathbf{k}=(11,12,10,12,10,10)$. We will compute $\Theta_F$ ($\S$\ref{['example3/3']}) in this case to be $(u-1)^3(u^2-(t_1+t_1t_2+7+1/t_1+1/{t_1t_2})u+1)$.
  • Figure 3: A modification of the train track $\tau$ in a neighborhood of the only vertex $V$ of $\tau$, such that the modified train track $\tau'$ is trivalent.

Theorems & Definitions (14)

  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Definition 1
  • proof : Proof of Theorem \ref{['mainThm']}
  • Remark 1
  • Remark 2
  • Lemma 4
  • proof
  • Proposition 5
  • ...and 4 more