Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2401.06870 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913213802610688 |
|---|---|
| author | Dolgushev, Vasily A. Guynee, Jacob J. |
| author_facet | Dolgushev, Vasily A. Guynee, Jacob J. |
| contents | Let $B_3$ be the Artin braid group on 3 strands and $PB_3$ be the corresponding pure braid group. In this paper, we construct the groupoid $GTSh$ of GT-shadows for a (possibly more tractable) version $GT_0$ of the Grothendieck-Teichmueller group $GT$ introduced by D. Harbater and L. Schneps in 2000. We call this group the gentle version of $GT$ and denote it by $GT_{gen}$. The objects of $GTSh$ are finite index normal subgroups $N$ of $B_3$ satisfying the condition $N \subset PB_3$. Morphisms of $GTSh$ are called GT-shadows and they may be thought of as approximations to elements of $GT_{gen}$. We show how GT-shadows can be obtained from elements of $GT_{gen}$ and prove that $GT_{gen}$ is isomorphic to the limit of a certain functor defined in terms of the groupoid $GTSh$. Using this result, we get a criterion for identifying genuine GT-shadows. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06870 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | GT-shadows for the gentle version of the Grothendieck-Teichmueller group Dolgushev, Vasily A. Guynee, Jacob J. Group Theory Number Theory Let $B_3$ be the Artin braid group on 3 strands and $PB_3$ be the corresponding pure braid group. In this paper, we construct the groupoid $GTSh$ of GT-shadows for a (possibly more tractable) version $GT_0$ of the Grothendieck-Teichmueller group $GT$ introduced by D. Harbater and L. Schneps in 2000. We call this group the gentle version of $GT$ and denote it by $GT_{gen}$. The objects of $GTSh$ are finite index normal subgroups $N$ of $B_3$ satisfying the condition $N \subset PB_3$. Morphisms of $GTSh$ are called GT-shadows and they may be thought of as approximations to elements of $GT_{gen}$. We show how GT-shadows can be obtained from elements of $GT_{gen}$ and prove that $GT_{gen}$ is isomorphic to the limit of a certain functor defined in terms of the groupoid $GTSh$. Using this result, we get a criterion for identifying genuine GT-shadows. |
| title | GT-shadows for the gentle version of the Grothendieck-Teichmueller group |
| topic | Group Theory Number Theory |
| url | https://arxiv.org/abs/2401.06870 |