Geometric subgroups of mapping class groups

This paper is a study of the subgroups of mapping class groups of Riemann s urfaces, called "geometric" subgroups, corresponding to the inclusion of su b-surfaces. Our analysis includes surfaces with boundary and with punctures . The centres of all the mapping class groups are calculated. We determine the kernel of inclusion-induced maps of the mapping class group of a subsur face, and give necessary and sufficient conditions for injectivity. In the injective case, we show that the commensurability class of a geometric subg roup completely determines up to isotopy the defining subsurface, and we ch aracterize centralizers, normalizers, and commensurators of geometric subgr oups.