Mapping class groups (Q2776340)

Mapping class groups of \(2\)-manifolds have been investigated for nearly 100 years, and now enjoy a vast literature representing diverse mathematical viewpoints. This lengthy survey article treats a considerable portion of the classical and modern theory, with particular attention to the role of the complex of curves. This is the simplicial complex \(C(S)\) whose vertices are the isotopy classes of simple closed loops in the connected \(2\)-manifold \(S\), and whose simplices are spanned by collections of vertices for which there exist pairwise disjoint representatives. Although \(C(S)\) is not locally finite, it is finite-dimensional. NEWLINENEWLINENEWLINEAfter a general introduction to the topology and geometry of surfaces, including the classical Dehn-Nielsen-Baer theorems, the author examines the homotopy type of \(C(S)\). Apart from a few small-genus exceptions, it is homotopy equivalent to a one-point union of spheres of the same dimension, this dimension being determined by a simple formula involving the genus and number of boundary components of \(S\). This was first proven by J. Harer, using Teichmüller theory. The author gives a fairly complete exposition of his own proof using Morse-Cerf theory. A discussion of recent results of H. Masur and Y. Minsky about the \(\delta\)-hyperbolicity of \(C(S)\) is also included. NEWLINENEWLINENEWLINEA proof is given of the fundamental fact that the mapping class group is generated by Dehn twists, in fact by an explicit finite collection of Dehn twists. The author's proof makes use of \(C(S)\), and is closer to Dehn's original approach than to Lickorish's later rediscovery. Some Teichmüller theory is developed, using the definition of Teichmüller space in terms of metrics of constant curvature, rather than in terms of Riemann surfaces. The author finds this viewpoint ``closer in spirit to that of Fricke than to the approach of Teichmüller himself.'' NEWLINENEWLINENEWLINEThe cohomology of mapping class groups receives considerable attention. Utilizing a quick development of the cohomology of groups due to Bieri and Eckmann, the author explains the landmark results about virtual cohomological dimension and homological stability of mapping class groups, and discusses recent work. A less extensive survey of the Nielsen-Thurston theory of surface diffeomorphisms is provided. NEWLINENEWLINENEWLINEThe author's important results on the automorphisms of \(C(S)\) are sketched in considerable detail, and applied to give a complete description of isomorphisms between subgroups of finite index in mapping class groups, and to prove that (apart from the usual small-genus exceptions) these subgroups have finite outer automorphism group. NEWLINENEWLINENEWLINEThe final chapter examines the considerable literature on comparison of mapping class groups with arithmetic groups. W. Harvey was the first to note that mapping class groups seem closely related to arithmetic groups, and the author was the first to prove that they are not actually arithmetic. He includes here two proofs, one using deep properties of arithmetic groups and relatively easy observations about mapping class groups, and the other using more basic theory of arithmetic groups but deeper properties of mapping class groups. Although mapping class groups are not arithmetic, the analogy is quite strong and has stimulated considerable research. The author sketches recent work showing that mapping class groups are of rank \(1\), in a sense that generalizes the rank of an arithmetic group, and closes with a very up-to-date discussion of current research. NEWLINENEWLINENEWLINEAlthough this very well-written survey necessarily leaves out some topics (for example, there is little about the Torelli group or group actions on surfaces), there is much here for both the novice and the expert.NEWLINENEWLINEFor the entire collection see [Zbl 0977.00029].