A brief introduction to mapping class groups (Q2848311)

This survey article summarizes the contents of a set of lectures delivered by the author at the Park City Mathematics Institute program on moduli spaces of Riemann surfaces in July, 2011. With regard to the fact that the (analytic) moduli space \(M_{g,n}\) of \(n\)-pointed compact Riemann surfaces of genus \(g\) can be constructed via Teichmüller theory by using the so-called mapping class group \(\text{Mod}(S)\) of a fixed surface \(S:= S_{g,n}\) of such type, the aim of these lectures was to give an introduction to mapping class groups, discuss their basic topological properties, and explain the Nielsen-Thurston classification of mapping classes to graduate students and researchers interested in these topics.NEWLINENEWLINE As the author points out in the introduction to his notes, all of this beautiful material is meanwhile classic, and is thoroughly covered in a number of books and articles suitable for complementary, more detailed reading.NEWLINENEWLINE Especially the most recent textbook ``A primer on mapping class groups'' by \textit{B. Farb} and \textit{D. Margalit} [Princeton Math. Series. Princeton, NJ: Princeton Univ. Press (2011; Zbl 1245.57002)] may be regarded as a perfect reference for the subject discussed in the article under review.NEWLINENEWLINE The first section provides the basic definitions, examples, and structural properties of mapping class groups, thereby emphasizing the distinguished role of Dehn twists as generators. Section 2 explains the connection with hyperbolic geometry, laminations, foliations, and pseudo-Anosov maps in the context of surfaces, while Section 3 is devoted to the Nielsen-Thurston classification theorem for mapping classes \(f\in \text{Mod}(S)\). The proof of the latter is vividly sketched along the approach of \textit{A. J. Casson} and \textit{S. A. Bleiler}, as pursued in their book ``Automorphisms of surfaces after Nielsen and Thurston'' [London Mathematical Society Student Texts, 9. Cambridge (UK) etc.: Cambridge University Press. 105 p. (1988; Zbl 0649.57008)]. In Section 4, canonical reducing systems for mapping classes are introduced to accomplish the last remaining step in the proof of the Nielsen-Thurston classification theorem, on the one hand, and some consequences of the entire discussion are depicted on the other. Finally, Section 5 gives a brief overview of parts of the related literature, thereby providing hints to further reading and recent developments. The rich bibliography of 75 references enhances the present survey on mapping class groups very effectively, and the many beautiful pictures illustrating the abstract ideas throughout the text are just as helpful for beginners in the field.NEWLINENEWLINEFor the entire collection see [Zbl 1272.30002].