Local wild mapping class groups and cabled braids
[Groupes modulaires sauvages locaux et tresses câblées]
Douçot, Jean1 ; Rembado, Gabriele2 ; Tamiozzo, Matteo3
1Department of Mathematics, University of Lisbon, Campo Grande, Edifício C6, PT-1749-016 Lisboa (Portugal)
2Hausdorff Centre for Mathematics, University of Bonn, 60 Endenicher Allee, D-53115 Bonn (Germany)
3Department of Mathematics, University of Warwick, Zeeman Building, CV47AL Coventry (UK)
Annales de l'Institut Fourier, Online first, 54 p.

We define and study some generalisations of pure $\mathfrak{g}$-braid groups, for any complex reductive Lie algebra $\mathfrak{g}$. They naturally occur in the theory of isomonodromic deformations for meromorphic connections with irregular singularities on principal bundles over Riemann surfaces, covering the general untwisted case, going beyond the case of generic irregular types. These generalised braid groups make up local pieces of the wild mapping class groups, which in turn extend the usual mapping class groups and govern the braiding of Stokes data.

We establish a general product decomposition for these local wild mapping class groups, and in all classical cases define a fission tree governing the decomposition; in particular in type $D$ we will find a factor which is not isomorphic to any pure braid group coming from a root system. In type $A$, the fission tree and the pure braid group operad yields a proof of the corresponding “multi-scale” braiding conjecture.

Nous définissons et étudions quelques généralisations des groupes de $\mathfrak{g}$-tresses purs, pour toute algèbre de Lie réductive complexe $\mathfrak{g}$. Ils apparaissent naturellement dans la théorie des déformations isomonodromiques pour les connexions méromorphes à singularités irrégulières sur les fibrés principaux au-dessus des surfaces de Riemann, couvrant le cas général non twisté, et allant au-delà du cas des types irréguliers génériques. Ces groupes de tresses généralisés constituent des parties locales des groupes modulaires sauvages, qui à leur tour étendent les groupes modulaires habituels et contrôlent le tressage des données de Stokes.

Nous établissons une décomposition générale en produit pour ces groupes modulaires sauvages locaux et, dans tous les cas classiques, nous définissons un arbre de fission qui contrôle cette décomposition. En particulier dans le type $D$ nous trouvons un facteur qui n’est pas isomorphe à un groupe de tresses pur provenant d’un systeme de racines. Dans le type $A$, l’arbre de fission et l’opérade du groupe de tresses pur donne une preuve de la conjecture de tressage « multi-échelle » correspondante.

Reçu le :
Révisé le :
Accepté le :
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DOI : 10.5802/aif.3778
Classification : 14D22, 14F35, 17B22, 32S22
Keywords: Braid groups, braid group operads, isomonodromic deformations, root systems, reflection groups, hyperplane arrangements, fundamental groups, moduli spaces

Douçot, Jean  1   ; Rembado, Gabriele  2   ; Tamiozzo, Matteo  3

1 Department of Mathematics, University of Lisbon, Campo Grande, Edifício C6, PT-1749-016 Lisboa (Portugal)
2 Hausdorff Centre for Mathematics, University of Bonn, 60 Endenicher Allee, D-53115 Bonn (Germany)
3 Department of Mathematics, University of Warwick, Zeeman Building, CV47AL Coventry (UK) 
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Douçot, Jean; Rembado, Gabriele; Tamiozzo, Matteo. Local wild mapping class groups and cabled braids. Annales de l'Institut Fourier, Online first, 54 p.

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