Introduction to arithmetic mapping class groups (Q2848319)

This survey article contains the lecture notes for the author's mini-course taught at the Graduate Summer School of the IAS/Park City Mathematics Institute in July 2011. As the author points out, the main goal of his five lectures is to explain why the absolute Galois group \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) and its action on the profinite completion of the fundamental group of a \(n\)-pointed Riemann surface of genus \(g\) appear quite naturally in the study of the corresponding mapping class group, and therefore also in the geometric description of the moduli space \(M_{g,n}\) of such surfaces.NEWLINENEWLINE This approach is based on Grothendieck's theory of the algebraic fundamental group of an algebraic variety, and consequently this abstract framework is briefly introduced in Lecture 1. In this context, the algebraic fundamental group of an algebraic variety \(K\) defined over a subfield \(K\) of \(\mathbb{C}\) appears as the profinite completion of the classic topological fundamental group, and its connection with the Galois group \(\mathrm{Gal}(\overline K/K)\) is explained via Grothendieck's seminal work developed in the famous seminar notes ``SGA 1'' [Lecture Notes in Mathematics. 224. Berlin-Heidelberg-New York: Springer-Verlag. XXII, 447 p. (1971; Zbl 0234.14002)]. In the special case of the Teichmüller description of the moduli space \(M_{g,n}(\mathbb{C})\) it is shown that the Galois group \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) naturally appears in a short exact sequence containing the algebraic fundamental groups of \(M_{g,n}(\mathbb{C})\) and \(M_{g,n}(\mathbb{Q})\), on the one hand, and in an exact sequence containing the so-called arithmetic mapping class group \(\mathrm{AMCG}_{g,n}\) and the profinite completion of the ordinary mapping class group \(\mathrm{MCG}_{g,n}\) alternatively.NEWLINENEWLINE Lecture 2 briefly recalls the monodromy representation on topological fundamental groups, thereby pointing out that the above Galois representation of \(\mathrm{Gal}(\mathbb{Q}/\mathbb{Q})\) may be viewed as an analogue of the latter in the arithmetic/algebraic case.NEWLINENEWLINE Lecture 3 provides a sketchy description of the arithmetic mapping class groups \(\mathrm{AWCG}_{g,n}\), with the main reference being the related work of \textit{T. Oda} [Étale homotopy type of the moduli spaces of algebraic curves. Geometric Galois actions 1. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 242, 85--95 (1997; Zbl 0902.14019)].NEWLINENEWLINE Lecture 4 points to some more recent results concerning the properties of various Galois representations of \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) with regard to outer fundamental groups of \((g,n)\)-surfaces, or to the profinite completion of the mapping class group \(\mathrm{MCG}_{g,n}\), respectively. One of the main references, in this context, is the author's own work [\textit{M. Matsumoto} and \textit{A. Tamagawa}, Am. J. Math. 122, No. 5, 1017--1026 (2000; Zbl 0993.12002)].NEWLINENEWLINE Finally, Lecture 5 discusses the question of how these Galois actions of \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) can be used to obtain a bound for the size of the mapping class group \(\mathrm{MCG}_{g,n}\) in the group of automorphisms of the Lie algebraization of the topological fundamental group of a \((g,n)\)-surface by giving Galois obstructions to the surjectivity of the so-called higher Johnson homomorpbisms. In this context, the geometry of the mixed Hodge structure on the topological fundamental group of a \((g,n)\)-surface plays a crucial role, just as the recent progress concerning related conjectures of Deligne-Ihara and of T. Oda does. The paper ends with an appendix, the purpose of which is to describe algebraic fundamental groups in terms of étale coverings and fiber functors à la A. Grothendieck.NEWLINENEWLINEFor the entire collection see [Zbl 1272.30002].