The Dehn twist coefficient for big and small mapping class groups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916839059095552 |
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| author | Feller, Peter Hubbard, Diana Turner, Hannah |
| author_facet | Feller, Peter Hubbard, Diana Turner, Hannah |
| contents | We study a quasimorphism, which we call the Dehn twist coefficient (DTC), from the mapping class group of a surface (with a chosen compact boundary component) that generalizes the well-studied fractional Dehn twist coefficient (FDTC) to surfaces of infinite type. Indeed, for surfaces of finite type the DTC coincides with the FDTC. We provide a characterization of the DTC as the unique homogeneous quasimorphism satisfying certain positivity conditions. This characterization is new even for the classical finite-type case and requires minimal input beyond elementary topology.
The FDTC has image contained in $\mathbb{Q}$. In contrast to this, we find that for some surfaces of infinite type the DTC has image all of $\mathbb{R}$. To see this we provide a new construction of maps with irrational rotation behavior for some surfaces of infinite type with a countable space of ends or even just one end. In fact, we find that the DTC is the right tool to detect irrational rotation behavior, even for surfaces without boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_06214 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Dehn twist coefficient for big and small mapping class groups Feller, Peter Hubbard, Diana Turner, Hannah Geometric Topology 57K20, 20F36, 20F60 We study a quasimorphism, which we call the Dehn twist coefficient (DTC), from the mapping class group of a surface (with a chosen compact boundary component) that generalizes the well-studied fractional Dehn twist coefficient (FDTC) to surfaces of infinite type. Indeed, for surfaces of finite type the DTC coincides with the FDTC. We provide a characterization of the DTC as the unique homogeneous quasimorphism satisfying certain positivity conditions. This characterization is new even for the classical finite-type case and requires minimal input beyond elementary topology. The FDTC has image contained in $\mathbb{Q}$. In contrast to this, we find that for some surfaces of infinite type the DTC has image all of $\mathbb{R}$. To see this we provide a new construction of maps with irrational rotation behavior for some surfaces of infinite type with a countable space of ends or even just one end. In fact, we find that the DTC is the right tool to detect irrational rotation behavior, even for surfaces without boundary. |
| title | The Dehn twist coefficient for big and small mapping class groups |
| topic | Geometric Topology 57K20, 20F36, 20F60 |
| url | https://arxiv.org/abs/2308.06214 |