New York Journal of Mathematics
Volume 23 (2017) 671-697


Leah R. Childers

The automorphism group of the hyperelliptic Torelli group

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Published: June 8, 2017
Keywords: Torelli group, hyperelliptic Torelli group, symmetric mapping class group, Johnson kernel, Dehn twist
Subject: Primary: 57S05. Secondary: 20F36, 57M07

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. The main result of this paper is that, for g ≧ 3, we have
Aut(SI(Sg)) ≅ SMod±(Sg)/<ι>,
where SMod±(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±(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.

Author information

Mathematics Department, Pittsburg State University, 1701 S. Broadway, Pittsburg, KS 66762