About the even minimal stratum of translation surfaces in genus 4
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909554745278464 |
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| author | Giannini, Riccardo |
| author_facet | Giannini, Riccardo |
| contents | In the present note, we complete the correspondence between stratum components of translation surfaces in low genus and finite-type Artin groups with defining Dynkin diagram containing $E_6$. In an earlier work, we showed that in genus $3$ the monodromy of the non-hyperelliptic connected components $\mathcal{H}^{\operatorname{odd}}(4)$ and $\mathcal{H}(3,1)$ are highly non-injective, as the respective kernels contain a non-abelian free group of rank $2$. The result holds since both the stratum components are orbifold classifying spaces for central extensions of the inner automorphism groups of the finite-type Artin groups $A_{E_6}$ and $A_{E_7}$, respectively. The following is a note extending the same result to the stratum $\mathcal{H}^{\operatorname{even}}(6)$ in genus $4$, which is an orbifold classifying space for a central extension of the group $\operatorname{Inn}(A_{E_8})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_13936 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | About the even minimal stratum of translation surfaces in genus 4 Giannini, Riccardo Geometric Topology Algebraic Geometry Group Theory In the present note, we complete the correspondence between stratum components of translation surfaces in low genus and finite-type Artin groups with defining Dynkin diagram containing $E_6$. In an earlier work, we showed that in genus $3$ the monodromy of the non-hyperelliptic connected components $\mathcal{H}^{\operatorname{odd}}(4)$ and $\mathcal{H}(3,1)$ are highly non-injective, as the respective kernels contain a non-abelian free group of rank $2$. The result holds since both the stratum components are orbifold classifying spaces for central extensions of the inner automorphism groups of the finite-type Artin groups $A_{E_6}$ and $A_{E_7}$, respectively. The following is a note extending the same result to the stratum $\mathcal{H}^{\operatorname{even}}(6)$ in genus $4$, which is an orbifold classifying space for a central extension of the group $\operatorname{Inn}(A_{E_8})$. |
| title | About the even minimal stratum of translation surfaces in genus 4 |
| topic | Geometric Topology Algebraic Geometry Group Theory |
| url | https://arxiv.org/abs/2412.13936 |