Action of the Grothendieck-Teichmüller group on torsion elements of full Teichmüller modular groups in genus zero
This paper concerns the action of the Grothendieck-Teichmüller group \(\widehat{GT}\) on torsion elements of the profinite fundamental group \(\pi_1^{\mathrm{geom}}(\mathcal{M}_{0,[n]})\) of the moduli space of genus zero curves with \(n\) unordered marked points. This latter fundamental group is isomorphic to the profinite completion \(\hat{\Gamma}_{0,[n]}\) of the full mapping class group \(\Gamma_{0,[n]}\), consisting of oriented diffeomorphisms of the \(n\)-punctured genus 0 surface modulo those isotopic to the identity. Since the discrete group \(\Gamma_{0,[n]}\) is residually finite, it injects into its profinite completion, and the image of the torsion is called the \textit{geometric torsion} of \(\hat{\Gamma}_{0,[n]}\). The first main result of the paper (Theorem A) states that every prime order torsion element of \(\hat{\Gamma}_{0,[n]}\) is conjugate to a geometric torsion element. The approach is cohomological, making use of the fact that the mapping class groups are good in the sense of Serre. The restriction to the prime order case results from some complications due to the fact that \(\Gamma_{0,[n]}\) does not satisfy a certain property \((\star)\) concerning conjugacy classes of maximal finite subgroups (Example 3.5). Theorem A allows the author to use the explicit action of \(\widehat{GT}\) on the braid group generators of \(\hat{\Gamma}_{0,[n]}\) in order to prove Theorem B: \(\widehat{GT}\) acts on the prime order torsion of \(\hat{\Gamma}_{0,[n]}\) by \(\lambda\)-conjugacy. In particular, since \(G_\mathbb{Q}\hookrightarrow \widehat{GT}\), this result recovers (for prime order torsion) the author's earlier result that the \(G_\mathbb{Q}\) action on geometric torsion is given by \(\chi\)-conjugacy, where \(\chi\) is the cyclotomic character.
scientific article; zbMATH DE number 6142426