On the surjectivity of the Symplectic representation of the mapping class group
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866915560658305024 |
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| author | Baik, Hyungryul Choi, Inhyeok Kim, Dongryul M. |
| author_facet | Baik, Hyungryul Choi, Inhyeok Kim, Dongryul M. |
| contents | In this note, we study the symplectic representation of the mapping class group. In particular, we discuss the surjecivity of the representation restricted to certain mapping classes. It is well-known that the representation itself is surjective. In fact the representation is still surjective after restricting on pseudo-Anosov mapping classes. However, we show that the surjectivity does not hold when the representation is restricted on orientable pseudo-Anosovs, even after reducing its codomain to integer symplectic matrices with a bi-Perron leading eigenvalue. In order to prove the non-surjectivity, we explicitly construct an infinite family of symplectic matrices with a bi-Perron leading eigenvalue which cannot be obtained as the symplectic representation of an orientable pseudo-Anosov mapping class. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_10142 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On the surjectivity of the Symplectic representation of the mapping class group Baik, Hyungryul Choi, Inhyeok Kim, Dongryul M. Geometric Topology Group Theory In this note, we study the symplectic representation of the mapping class group. In particular, we discuss the surjecivity of the representation restricted to certain mapping classes. It is well-known that the representation itself is surjective. In fact the representation is still surjective after restricting on pseudo-Anosov mapping classes. However, we show that the surjectivity does not hold when the representation is restricted on orientable pseudo-Anosovs, even after reducing its codomain to integer symplectic matrices with a bi-Perron leading eigenvalue. In order to prove the non-surjectivity, we explicitly construct an infinite family of symplectic matrices with a bi-Perron leading eigenvalue which cannot be obtained as the symplectic representation of an orientable pseudo-Anosov mapping class. |
| title | On the surjectivity of the Symplectic representation of the mapping class group |
| topic | Geometric Topology Group Theory |
| url | https://arxiv.org/abs/2008.10142 |