The index of the normal closure of a power of a twist in the mapping class group

Charalampos Stylianakis (University of Glasgow)

Wednesday 2nd December, 2015 16:00-17:00 Maths 522

Abstract

The mapping class group of an orientable surface is the group of homotopy classes of homeomorphisms that preserve the orientation of the surface and fix the boundary pointwise. It is a classical result that Dehn twists generate the mapping class group. We consider a subgroup of the mapping class group, the hyperelliptic mapping class group, that is, the group consisting of mapping classes that commute with a fixed hyperelliptic involution. In this talk we will investigate subgroups of hyperelliptic mapping class groups, namely, the subgroups generated by powers of Dehn twists. It can be shown that in most cases the subgroups above have infinite index in the hyperelliptic mapping class group. The tool we use to prove the latter result is a linear representation that factors through the Hecke algebra, namely the Jones representation of the hyperelliptic mapping class group.

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