Galois actions on surfaces and a higher genus Grothendieck-Teichmüller group

Abstract

We construct a modular operad S of mapping class groupoids modelling the higher-genus Teichmüller tower, with compositions and contractions encoding surface gluing. We prove a presentation theorem for maps out of S using genus-zero and genus-one generators and relations. Applying the work of Nakamura--Schneps, we obtain a faithful action of their subgroup Γ̂⊆ GT̂ on Ŝ, and hence a faithful Galois action. The genus-zero truncation identifies with the cyclic operad of parenthesized ribbon braids, whose object-fixing profinite automorphism group is GT̂. The completed classifying spaces form a modular ∞-operad whose corolla values are geometric étale homotopy types of moduli stacks of curves with nonzero tangent vectors at their marked points. The Γ̂-action induces a faithful action by homotopy automorphisms on the associated tower of profinite homotopy $1$-types, which agrees with the étale model in genus zero and one.

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BibTeX RIS

BibTeX

@misc{bonatto2026galois,
  title = {{Galois actions on surfaces and a higher genus Grothendieck-Teichmüller group}},
  author = {Luciana Basualdo Bonatto and Marcy Robertson},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.01466},
  eprint = {2606.01466},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.01466},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Galois actions on surfaces and a higher genus Grothendieck-Teichmüller group
AU  - Luciana Basualdo Bonatto
AU  - Marcy Robertson
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2606.01466
UR  - https://arxiv.org/abs/2606.01466
AB  - We construct a modular operad S of mapping class groupoids modelling the higher-genus Teichmüller tower, with compositions and contractions encoding surface gluing. We prove a presentation theorem for maps out of S using genus-zero and genus-one generators and relations. Applying the work of Nakamura--Schneps, we obtain a faithful action of their subgroup Γ̂⊆ GT̂ on Ŝ, and hence a faithful Galois action. The genus-zero truncation identifies with the cyclic operad of parenthesized ribbon braids, whose object-fixing profinite automorphism group is GT̂. The completed classifying spaces form a modular ∞-operad whose corolla values are geometric étale homotopy types of moduli stacks of curves with nonzero tangent vectors at their marked points. The Γ̂-action induces a faithful action by homotopy automorphisms on the associated tower of profinite homotopy $1$-types, which agrees with the étale model in genus zero and one.
ER  -

Abstract

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