Simplicial actions of mapping class groups

Abstract

In this chapter, we review the actions of the (extended) mapping class group of a surface of finite type by simplicial automorphisms on various abstract simplicial complexes (the curve complex, the arc complex, the cut system complex, the pants decomposition complex, the ideal triangulation complex, etc.). Each of these complexes has some special combinatorial features, and there are interesting questions that are particular to each one. We study in detail the actions on recently defined complexes, namely, the complex of domains and some of its subcomplexes. In most cases mentioned, the automorphism group of the simplicial automorphism coincides with the (injective) image of the mapping class group in that automorphism group. In the case of the complex of domains, this does not hold, as soon as the surface has at least two boundary components. We compute the automorphism group of the complex of domains, and we introduce a subcomplex of this complex, the truncated complex of domains, whose automorphism group is the extended mapping class group. The survey contains expository sections on surfaces and related objects and on simplicial complexes.