Galois actions on surfaces and a higher genus Grothendieck-Teichmüller group
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arXiv
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| Format: | Preprint |
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2026
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| author | Bonatto, Luciana Basualdo Robertson, Marcy |
| author_facet | Bonatto, Luciana Basualdo Robertson, Marcy |
| contents | We construct an operadic model for the higher-genus Teichmüller tower. More precisely, we define a modular operad $\mathbf{S}$ in groupoids built from mapping class groups, with compositions and contractions encoding gluing operations on surfaces. We prove a presentation theorem for maps out of $\mathbf{S}$, showing that they are determined by a small number of genus-zero and genus-one generators and relations. Using this presentation and the work of Nakamura--Schneps, we construct a faithful action of the Nakamura--Schneps subgroup $\widehatΓ\subseteq\widehat{\mathsf{GT}}$ on the profinite completion $\widehat{\mathbf{S}}$, and hence an action of $\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)$. The genus-zero truncation of $\mathbf{S}$ recovers the cyclic operad of parenthesized ribbon braids, and its group of object-fixing profinite automorphisms recovers $\widehat{\mathsf{GT}}$. Finally, the profinite completion of the classifying spaces of $\mathbf{S}$ assemble into a modular $\infty$-operad in profinite spaces whose values identify with the étale homotopy types of moduli stacks of curves with marked tangent vectors, and the $\widehatΓ$-action extends to this homotopy-coherent Teichmüller tower. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01466 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Galois actions on surfaces and a higher genus Grothendieck-Teichmüller group Bonatto, Luciana Basualdo Robertson, Marcy Algebraic Topology Category Theory Quantum Algebra 18M75, 18N70 We construct an operadic model for the higher-genus Teichmüller tower. More precisely, we define a modular operad $\mathbf{S}$ in groupoids built from mapping class groups, with compositions and contractions encoding gluing operations on surfaces. We prove a presentation theorem for maps out of $\mathbf{S}$, showing that they are determined by a small number of genus-zero and genus-one generators and relations. Using this presentation and the work of Nakamura--Schneps, we construct a faithful action of the Nakamura--Schneps subgroup $\widehatΓ\subseteq\widehat{\mathsf{GT}}$ on the profinite completion $\widehat{\mathbf{S}}$, and hence an action of $\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)$. The genus-zero truncation of $\mathbf{S}$ recovers the cyclic operad of parenthesized ribbon braids, and its group of object-fixing profinite automorphisms recovers $\widehat{\mathsf{GT}}$. Finally, the profinite completion of the classifying spaces of $\mathbf{S}$ assemble into a modular $\infty$-operad in profinite spaces whose values identify with the étale homotopy types of moduli stacks of curves with marked tangent vectors, and the $\widehatΓ$-action extends to this homotopy-coherent Teichmüller tower. |
| title | Galois actions on surfaces and a higher genus Grothendieck-Teichmüller group |
| topic | Algebraic Topology Category Theory Quantum Algebra 18M75, 18N70 |
| url | https://arxiv.org/abs/2606.01466 |