Dans cet article, nous caractérisons l'action du groupe de Galois absolu sur les groupes d'inertie champêtre géométriques cycliques et sans factorisation étale du groupe fondamental géométrique des espaces de modules de courbes marquées. Nous établissons par ailleurs la même action sur les éléments de torsion profinis d'ordre premier en genre 2.

In this paper we define a new version of Grothendieck–Teichmüller group [Formula: see text] defined by three generalized equations coming from finite-order diffeomorphisms, and we prove that it is isomorphic to the known version IΓ of the Grothendieck–Teichmüller defined in [H. Nakamura and L. Schneps, On a subgroup of the Grothendieck–Teichmüller group acting on the tower of profinite Teichmüller modular groups, Invent. Math.141 (2000) 503–560]. We show that [Formula: see text] acts on the full mapping class groups [Formula: see text] for 2g - 2 + n > 0. We then prove that the conjugacy classes of prime-order torsion of [Formula: see text] are exactly the discrete prime-order ones of [Formula: see text]. Using this we prove that [Formula: see text] acts on prime-order torsion elements of [Formula: see text] in a particular way called λ-conjugacy, analogous to the Galois action on inertia.

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The geometric study of the absolute Galois group of the rational numbers has been a highly active research topic since the first milestones: Hilbert’s Irreducibility Theorem, Noether’s program, Riemann’s Existence Theorem. It gained special interest in the last decades with Grothendieck’s “Esquisse d’un programme”, his “Letter to Faltings” and Fried’s introduction of Hurwitz spaces. It grew on and thrived on a wide range of areas, e.g. formal algebraic geometry, Diophantine geometry, group theory. The recent years have seen the development and integration in algebraic geometry and Galois theory of new advanced techniques from algebraic stacks, \ell -adic representations and homotopy theories. It was the goal of this mini-workshop, to bring together an international panel of young and senior experts to draw bridges towards these fields of research and to incorporate new methods, techniques and structures in the development of geometric Galois theory.

We study Tannakian properties of the convolution product of perverse sheaves on elliptic curves. We establish that for certain sheaves with unipotent local monodromy over seven points the corresponding Tannaka group is isomorphic to G 2 {G_{2}} . This monodromy approach generalizes a result of Katz on the existence of G 2 {G_{2}} -motives in the middle cohomology of deformations of Beauville surfaces.