Year: 2025
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Rafiqi, Ahmad
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4417515103
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