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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.

We prove that the cohomological dimension of the Torelli group for a closed, connected, orientable surface of genus g \\geq 2 is equal to 3g-5. This answers a question of Mess, who proved the lower bound and settled the case of 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 2g-3. For 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.

The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. Putman and the authors proved that this group is generated by Dehn twists about separating curves fixed by the hyperelliptic involution. In this paper, we introduce an algorithmic approach to factoring a wide class of elements of the hyperelliptic Torelli group into such Dehn twists, and apply our methods to several basic types of elements. As one consequence, we answer an old question of Dennis Johnson.

We give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to the generating set consisting of all simple closed curves.

The theme of this paper is that algebraic complexity implies dynamical complexity for pseudo-Anosov homeomorphisms of a closed surface${\\rm{S}}_{\\rm{g}}$of genus g. Penner proved that the logarithm of the minimal dilatation for a pseudo-Anosov homeomorphism of${\\rm{S}}_{\\rm{g}}$tends to zero at the rate 1/g. We consider here the smallest dilatation of any pseudo-Anosov homeomorphism of${\\rm{S}}_{\\rm{g}}$acting trivially on$\\Gamma /\\Gamma _k $, the quotient of$\\Gamma \\, = \\,\\pi _1 (S_g )$by the${\\rm{K}}^{{\\rm{th}}}$term of its lower central series, k > 1. In contrast to Penner's asymptotics, we prove that this minimal dilatation is bounded above and below, independently of g, with bounds tending to infinity with k. For example, in the case of the Torelli group${\\rm{I(S}}_g )$, we prove that${\\rm{L(I(S}}_g ))$, the logarithm of the minimal dilatation in${\\rm{I(S}}_g )$, satisfies .197 <${\\rm{L(I(S}}_g ))$< 4.127. In contrast, we find pseudo-Anosov mapping classes acting trivially on$\\Gamma /\\Gamma _k $whose asymptotic translation lengths on the complex of curves tend to 0 as g → ∞.

We construct the first examples of normal subgroups of mapping class groups that are isomorphic to non-free right-angled Artin groups. Our construction also gives normal, non-free right-angled Artin subgroups of other groups, such as braid groups and pure braid groups, as well as many subgroups of the mapping class group, such as the Torelli subgroup. Our work recovers and generalizes the seminal result of Dahmani–Guirardel–Osin, which gives free, purely pseudo-Anosov normal subgroups of mapping class groups. We give two applications of our methods: (1) we produce an explicit proper normal subgroup of the mapping class group that is not contained in any level $m$ congruence subgroup and (2) we produce an explicit example of a pseudo-Anosov mapping class with the property that all of its even powers have free normal closure and its odd powers normally generate the entire mapping class group. The technical theorem at the heart of our work is a new version of the windmill apparatus of Dahmani–Guirardel–Osin, which is tailored to the setting of group actions on the projection complexes of Bestvina–Bromberg–Fujiwara.

We extend a theorem of Masur--Wolf which states that given a finite-area hyperbolic surface S, every isometry of the Teichmüller space for S with the Weil--Petersson metric is induced by an element of the mapping class group for S. Our argument handles the previously untreated cases of the four times-punctured sphere, the once-punctured torus, and the twice-punctured torus.

We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at t = - 1 and also the fundamental group of the branch locus of the period mapping, and so we obtain analogous generating sets for those. One application is that each component in Torelli space of the locus of hyperelliptic curves becomes simply connected when curves of compact type are added.

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at ... and also the fundamental group of the branch locus of the period mapping, and so we obtain analogous generating sets for those. One application is that each component in Torelli space of the locus of hyperelliptic curves becomes simply connected when curves of compact type are added.