On cofinite subgroups of mapping class groups

Details On cofinite subgroups of mapping class groups On cofinite subgroups of mapping class groups

2003-07-09

arxiv.org/abs/math/0307110v1

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

Mathematics

Geometric Topology

9 pages, 1 figure

Scientific paper

For any positive integer $n$, we exhibit a cofinite subgroup $\Gamma_n$ of the mapping class group of a surface of genus at most two such that $\Gamma_n$ admits an epimorphism onto a free group of rank $n$. We conclude that $H^1(\Gamma_n;\Z)$ 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 $\Gamma_n$ 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.

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