On cofinite subgroups of mapping class groups

2003/07/09 by Mustafa Korkmaz
Mathematics · #math.GT #math.GR

paper · pdf

published as Proceedings of 9th Gokova Geometry-Topology Conference,Turkish Journal of Mathematics 27, no:1. (2003), 115-123 · 9 pages, 1 figure

arxiv created 2003/07/09 · arxiv updated 2009/12/01

Abstract

For any positive integer n, we exhibit a cofinite subgroup Γ_n of the mapping class group of a surface of genus at most two such that Γ_n admits an epimorphism onto a free group of rank n. We conclude that H¹(Γ_n;\Z) has rank at least n and the dimension of the second bounded cohomology of each of these mapping class groups is the cardinality of the continuum. In the case of genus two, the groups Γ_n can be chosen not to contain the Torelli group. Similarly for hyperelliptic mapping class groups. We also exhibit an automorphism of a subgroup of finite index in the mapping class group of a sphere with four punctures (or a torus) such that it is not the restriction of an endomorphism of the whole group.

Related

• Cohomological structure of the mapping class group and beyond
• Problems, Questions, and Conjectures about Mapping Class Groups
• On endomorphisms of surface mapping class groups
• Separating subgroups of mapping class groups in homological\n representations
• Constraining mapping class group homomorphisms using finite subgroups