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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2505.06930 |
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| _version_ | 1866915282883182592 |
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| author | Rafiqi, Ahmad |
| author_facet | Rafiqi, Ahmad |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06930 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Simple Computation of Teichmüller Polynomials from Integer Permutations Rafiqi, Ahmad Geometric Topology Algebraic Topology 37E30, 05A05, 37B40 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. |
| title | A Simple Computation of Teichmüller Polynomials from Integer Permutations |
| topic | Geometric Topology Algebraic Topology 37E30, 05A05, 37B40 |
| url | https://arxiv.org/abs/2505.06930 |