Geometric subgroups of mapping class groups

1999/06/18 by L. Paris, Paris, L., Dale Rolfsen +2
Mathematics · #20F38 (Secondary) #57N05 (Primary) #Advanced Algebra and Geometry #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #math.GR #math.GT #msc:20F38 #msc:57N05

paper · pdf · doi:10.48550/arxiv.math/9906122

42 pages, 23 figures. See also http://math.u-bourgogne.fr/topolog/paris/index.html

arxiv created 1999/06/18 · openalex publication_date 1999/06/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper is a study of the subgroups of the mapping class groups of Riemann surfaces, called "geometric" subgroups, corresponding to the inclusion of subsurfaces. 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 subsurface, and give necessary and sufficient conditions for injectivity. In the injective case, we show that the commensurability class of a geometric subgroup completely determines up to isotopy the defining subsurface, and we characterize centralizers, normalizers, and commensurators of geometric subgroups.

Citations

• A finite presentation of the mapping class group of an oriented surface
• Curves on 2-manifolds and isotopies
• A finite set of generators for the homeotopy group of a 2-manifold

Related

• Geometric normal subgroups in mapping class groups of punctured surfaces
• Subgroups of the mapping class group from the geometrical viewpoint
• Geometric survey of subgroups of mapping class groups
• Problems, Questions, and Conjectures about Mapping Class Groups
• Injective homomorphisms of mapping class groups of non-orientable surfaces