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| Formato: | Preprint |
| Publicado: |
2021
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| Acceso en línea: | https://arxiv.org/abs/2106.06645 |
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| _version_ | 1866916324826939392 |
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| author | Dolgushev, Vasily A. |
| author_facet | Dolgushev, Vasily A. |
| contents | GT-shadows are tantalizing objects that can be thought of as approximations of elements of the mysterious Grothendieck-Teichmueller group $\widehat{GT}$ introduced by V. Drinfeld in 1990. GT-shadows form a groupoid GTSh whose objects are finite index subgroups of the pure braid group PB_4, that are normal in B_4. The goal of this paper is to describe the action of GT-shadows on Grothendieck's child's drawings and show that this action agrees with that of $\widehat{GT}$. We discuss the hierarchy of orbits of child's drawings with respect to the actions of GTSh, $\widehat{GT}$, and the absolute Galois group G_Q of rationals. We prove that the monodromy group and the passport of a child's drawing are invariant with respect to the action of the subgroupoid of charming GT-shadows. We use the action of GT-shadows on child's drawings to prove that every Abelian child's drawing admits a Belyi pair defined over rationals. Finally, we describe selected examples of non-Abelian child's drawings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_06645 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The Action of GT-Shadows on Child's Drawings Dolgushev, Vasily A. Algebraic Topology Number Theory GT-shadows are tantalizing objects that can be thought of as approximations of elements of the mysterious Grothendieck-Teichmueller group $\widehat{GT}$ introduced by V. Drinfeld in 1990. GT-shadows form a groupoid GTSh whose objects are finite index subgroups of the pure braid group PB_4, that are normal in B_4. The goal of this paper is to describe the action of GT-shadows on Grothendieck's child's drawings and show that this action agrees with that of $\widehat{GT}$. We discuss the hierarchy of orbits of child's drawings with respect to the actions of GTSh, $\widehat{GT}$, and the absolute Galois group G_Q of rationals. We prove that the monodromy group and the passport of a child's drawing are invariant with respect to the action of the subgroupoid of charming GT-shadows. We use the action of GT-shadows on child's drawings to prove that every Abelian child's drawing admits a Belyi pair defined over rationals. Finally, we describe selected examples of non-Abelian child's drawings. |
| title | The Action of GT-Shadows on Child's Drawings |
| topic | Algebraic Topology Number Theory |
| url | https://arxiv.org/abs/2106.06645 |