Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t =-1

• 成果类型: Article
• 署名作者: Brendle, Tara; Margalit, Dan; Putman, Andrew
• 署名单位: University System of Georgia; Georgia Institute of Technology; University of Glasgow; Rice University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-014-0537-9
• 发表日期: 2015
• 页码: 263-310
• 关键词: mapping class group; braid group; PRESENTATIONS; surface; COHOMOLOGY; HOMOLOGY; genus-2; THEOREM; roots; unity

摘要

We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at t = -1 and also the fundamental group of the branch locus of the period mapping, and so we obtain analogous generating sets for those. One application is that each component in Torelli space of the locus of hyperelliptic curves becomes simply connected when curves of compact type are added.