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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.05638 |
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| _version_ | 1866915928604672000 |
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| author | Kasahara, Yasushi |
| author_facet | Kasahara, Yasushi |
| contents | Under the natural action of the pure mapping class group of a surface of genus at least three, we show that any global fixed point in the low-dimensional deformation space of the surface group corresponds to the trivial representation. A key observation is that such a global fixed point gives rise to a linear representation of the pure mapping class group of the corresponding surface with a marked point. Our argument works directly on the deformation space, without assuming the semisimplicity of representations, and yields a short alternative proof of a special case of a theorem of Landesman and Litt with a slight improvement. We also discuss a possible extension of this approach from global fixed points to finite orbits of the mapping class group action. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_05638 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global fixed point in low-dimensional surface group deformation space Kasahara, Yasushi Geometric Topology 57K20 Under the natural action of the pure mapping class group of a surface of genus at least three, we show that any global fixed point in the low-dimensional deformation space of the surface group corresponds to the trivial representation. A key observation is that such a global fixed point gives rise to a linear representation of the pure mapping class group of the corresponding surface with a marked point. Our argument works directly on the deformation space, without assuming the semisimplicity of representations, and yields a short alternative proof of a special case of a theorem of Landesman and Litt with a slight improvement. We also discuss a possible extension of this approach from global fixed points to finite orbits of the mapping class group action. |
| title | Global fixed point in low-dimensional surface group deformation space |
| topic | Geometric Topology 57K20 |
| url | https://arxiv.org/abs/2510.05638 |