Automorphisms of curves and their role in Grothendieck–Teichmüller theory

Abstract

AbstractThe goal of this article is to consider the role played by finite‐order elements in the mapping class groups and special loci on moduli spaces, within the framework of Grothendieck–Teichmüller theory, and in particularly in the genus zero case. Quotienting topological surfaces by finite‐order automorphisms induces certain morphisms between moduli spaces; we consider the corresponding special homomorphisms between mapping class groups. In genus zero, these morphisms are always defined over ℚ, so that the canonical outer Galois action on profinite genus zero mapping class groups respects the induced homomorphisms. For simplicity, we consider only the subgroup $ \widehat {GT} ^1 _{0,0} $ of elements F = (λ , f ) ∈ $ \widehat {GT} $ with λ = 1 and conditions on the Kummer characters ρ 2(F ) = ρ 3(F ) = 0. We define a subgroup $ \widehat {GS} ^1 _{0,0} $ ⊂ $ \widehat {GT} ^1 _{0,0} $ by considering only elements of $ \widehat {GT} ^1 _{0,0} $ respecting these homomorphisms on the first two levels in genus zero. Our main result states that the subgroup $ \widehat {GS} ^1 _{0,0} $, which is thus defined using only properties occurring in genus zero, possesses many remarkable geometric Galois‐type properties not visibly satisfied by $ \widehat {GT} $ itself, the most striking of which is that it is also an automorphism group of the profinite mapping class groups in all genera. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)