Counting mapping classes by Nielsen-Thurston type
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866918407142637568 |
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| author | Dowdall, Spencer Masur, Howard |
| author_facet | Dowdall, Spencer Masur, Howard |
| contents | This paper concerns the lattice counting problem for the mapping class group of a surface $S$ acting on Teichmüller space with the Teichmüller metric. In that problem the goal is to count the number of mapping classes that send a given point $x$ into the ball of radius $R$ centered about another point $y$. For the action of the entire group, Athreya, Bufetov, Eskin and Mirzakhani have shown this quantity is asymptotic to $e^{hR}$, where $h$ is the dimension of the Teichmüller space. We refine the problem by considering the action various distinguished subsets of elements and counting these separately. For the set of finite-order elements, we show the associated count grows coarsely at the rate of $e^{hR/2}$, that is, with half the exponent. For the reducible elements, the associated count grows coarsely at the rate of $e^{(h-1)R}$. Finally, for the set of all multitwists, the coarse growth rate is also $e^{hR/2}$. To obtain these quantitative estimates, we introduce a new notion in Teichmüller geometry, called complexity length, which reflects some aspects of the negative curvature of curve complexes and also has applications to counting problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_03721 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Counting mapping classes by Nielsen-Thurston type Dowdall, Spencer Masur, Howard Geometric Topology Dynamical Systems This paper concerns the lattice counting problem for the mapping class group of a surface $S$ acting on Teichmüller space with the Teichmüller metric. In that problem the goal is to count the number of mapping classes that send a given point $x$ into the ball of radius $R$ centered about another point $y$. For the action of the entire group, Athreya, Bufetov, Eskin and Mirzakhani have shown this quantity is asymptotic to $e^{hR}$, where $h$ is the dimension of the Teichmüller space. We refine the problem by considering the action various distinguished subsets of elements and counting these separately. For the set of finite-order elements, we show the associated count grows coarsely at the rate of $e^{hR/2}$, that is, with half the exponent. For the reducible elements, the associated count grows coarsely at the rate of $e^{(h-1)R}$. Finally, for the set of all multitwists, the coarse growth rate is also $e^{hR/2}$. To obtain these quantitative estimates, we introduce a new notion in Teichmüller geometry, called complexity length, which reflects some aspects of the negative curvature of curve complexes and also has applications to counting problems. |
| title | Counting mapping classes by Nielsen-Thurston type |
| topic | Geometric Topology Dynamical Systems |
| url | https://arxiv.org/abs/2305.03721 |