Counting conjugacy classes of fully irreducibles: double exponential growth
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866916128712818688 |
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| author | Kapovich, Ilya Pfaff, Catherine |
| author_facet | Kapovich, Ilya Pfaff, Catherine |
| contents | Inspired by results of Eskin and Mirzakhani counting closed geodesics of length $\le L$ in the moduli space of a fixed closed surface, we consider a similar question in the $Out(F_r)$ setting. The Eskin-Mirzakhani result can be equivalently stated in terms of counting the number of conjugacy classes (within the mapping class group) of pseudo-Anosovs whose dilitations have natural logarithm $\le L$. Let $\mathfrak N_r(L)$ denote the number of $Out(F_r)$-conjugacy classes of fully irreducibles satisfying that the natural logarithm of their dilatation is $\le L$. We prove for $r\ge 3$ that as $L\to\infty$, the number $\mathfrak N_r(L)$ has double exponential (in $L$) lower and upper bounds. These bounds reveal behavior not present in the surface setting or in classical hyperbolic dynamical systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1801_07471 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Counting conjugacy classes of fully irreducibles: double exponential growth Kapovich, Ilya Pfaff, Catherine Group Theory Dynamical Systems Geometric Topology 20F65 (Primary), 57M, 37B, 37D (Secondary) Inspired by results of Eskin and Mirzakhani counting closed geodesics of length $\le L$ in the moduli space of a fixed closed surface, we consider a similar question in the $Out(F_r)$ setting. The Eskin-Mirzakhani result can be equivalently stated in terms of counting the number of conjugacy classes (within the mapping class group) of pseudo-Anosovs whose dilitations have natural logarithm $\le L$. Let $\mathfrak N_r(L)$ denote the number of $Out(F_r)$-conjugacy classes of fully irreducibles satisfying that the natural logarithm of their dilatation is $\le L$. We prove for $r\ge 3$ that as $L\to\infty$, the number $\mathfrak N_r(L)$ has double exponential (in $L$) lower and upper bounds. These bounds reveal behavior not present in the surface setting or in classical hyperbolic dynamical systems. |
| title | Counting conjugacy classes of fully irreducibles: double exponential growth |
| topic | Group Theory Dynamical Systems Geometric Topology 20F65 (Primary), 57M, 37B, 37D (Secondary) |
| url | https://arxiv.org/abs/1801.07471 |