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The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. It begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn–Nielsen–Baer–theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

Let K be the subgroup of the extended mapping class group, Mod(S), generated by Dehn twists about separating curves. Assuming that S is a closed, orientable surface of genus at least 4, we confirm a conjecture of Farb that Comm(K), Aut(K) and Mod(S) are all isomorphic. More generally, we show that any injection of a finite index subgroup of K into the Torelli group I of S is induced by a homeomorphism. In particular, this proves that K is co-Hopfian and is characteristic in I. Further, we recover the result of Farb and Ivanov that any injection of a finite index subgroup of I into I is induced by a homeomorphism. Our method is to reformulate these group theoretic statements in terms of maps of curve complexes.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper37.abs.htm

We prove that the cohomological dimension of the Torelli group for a closed, connected, orientable surface of genus g ≥ 2 g \geq 2 is equal to 3 g − 5 3g-5 . This answers a question of Mess, who proved the lower bound and settled the case of g = 2 g=2 . We also find the cohomological dimension of the Johnson kernel (the subgroup of the Torelli group generated by Dehn twists about separating curves) to be 2 g − 3 2g-3 . For g ≥ 2 g \geq 2 , we prove that the top dimensional homology of the Torelli group is infinitely generated. Finally, we give a new proof of the theorem of Mess that gives a precise description of the Torelli group in genus 2. The main tool is a new contractible complex, called the “complex of minimizing cycles”, on which the Torelli group acts.

In loving memory of my mother, Batya IntroductionIn the theory of mapping class groups, "curve complexes" assume a role similar to the one that buildings play in the theory of linear groups. Ivanov, Korkmaz, and Luo showed that the automorphism group of the curve complex for a surface is generally isomorphic to the extended mapping class group of the surface. In this paper, we show that the same is true for the pants complex.Throughout, S is an orientable surface whose Euler characteristic χ (S) is negative, while g,b denotes a surface of genus g with b boundary components. Also, Mod(S) means the extended mapping class group of S (the group of homotopy classes of self-homeomorphisms of S).The pants complex of S, denoted C P (S), has vertices representing pants decompositions of S, edges connecting vertices whose pants decompositions differ by an elementary move, and 2-cells representing certain relations between elementary moves (see Sec. 2). Its 1-skeleton C 1 P (S) is called the pants graph and was introduced by Hatcher and Thurston. We give a detailed definition of the pants complex in Section 2.Brock proved that C 1 P (S) models the Teichmüller space endowed with the Weil-Petersson metric, T W P (S), in that the spaces are quasi-isometric (see [1]). Our results further indicate that C 1 P (S) is the "right" combinatorial model for T W P (S), in that Aut C 1 P (S) (the group of simplicial automorphisms of C 1 P (S)) is shown to be Mod(S). This is in consonance with the result of Masur and Wolf that the isometry group of T W P (S) is Mod(S) (see [10]).There is a natural action of Mod(S) on C 1 P (S); we prove that all automorphisms of C 1 P (S) are induced by Mod(S). The results of this paper can be summarized as follows:Aut C P (S) ∼ = Aut C 1 P (S) ∼ = Mod(S) for most surfaces S.