University of Arkansas Topology Seminar: 2/23/2017

Speaker: Leah Childers

Title: The automorphism group of the hyperelliptic Torelli group

The hyperelliptic Torelli group, SI(Sg), is the subgroup of the mapping class group consisting of those elements that commute with a fixed hyperelliptic involution ι and act trivially on the homology of the surface Sg. The group SI(Sg) appears in a variety of contexts, e.g., as a kernel of a Burau representation and as the fundamental group of the branch locus of the period mapping on Torelli space. In this talk we will show, for g ≥ 3, we have Aut(SI(Sg)) ≅ SMod±1(Sg)/⟨ι⟩, where SMod±1(Sg) is the extended hyperelliptic mapping class group. Our main tool is the symmetric separating curve complex, Cssep(Sg), and we show that if g ≥ 3, Aut(Cssep(Sg)) ≅ SMod±1(Sg)/⟨ι⟩. Another key ingredient is an algebraic characterization of Dehn twists about symmetric separating curves. These results are analogous to results of Ivanov, Farb-Ivanov, and Brendle-Margalit for the mapping class group, the Torelli group, and the Johnson kernel.