Abstract
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.
Highlights
We find simple generating sets for three closely related groups: 1. the hyperelliptic Torelli group SIg, that is, the subgroup of the mapping class group consisting of elements that commute with some fixed hyperelliptic involution and that act trivially on the homology of the surface; 2. the fundamental group of Hg, the branch locus of the period mapping from Torelli space to the Siegel upper half-plane; and
3. the kernel of βn, the Burau representation of the braid group evaluated at t = −1
The group SIg, the space Hg, and the representation βn arise in many places in algebraic geometry, number theory, and topology; see, e.g., the work of A’Campo [2], Arnol’d [3], Band–Boyland [5], Funar–Kohno [18], Gambaudo– Ghys [19], Hain [20], Khovanov–Seidel [25], Magnus–Peluso [27], McMullen [31], Morifuji [33], Venkataramana [41], and Yu [43]
Summary
We find simple generating sets for three closely related groups: 1. the hyperelliptic Torelli group SIg, that is, the subgroup of the mapping class group consisting of elements that commute with some fixed hyperelliptic involution and that act trivially on the homology of the surface; 2. the fundamental group of Hg, the branch locus of the period mapping from Torelli space to the Siegel upper half-plane; and. Theorem C For n ≥ 1, the group BIn is generated by squares of Dehn twists about curves in Dn surrounding odd numbers of marked points. The first two authors proved [11, Theorem 1.2] that BI2g+2 is isomorphic to (BI2g+1/Z) F∞ and that each element of the F∞ subgroup is a product of squares of Dehn twists about curves surrounding odd numbers of marked points and so again by Mess’s theorem we obtain that BI6 is isomorphic to F∞ F∞ and that it satisfies Theorem C. 2g+1 is the group generated by squares of Dehn twists about curves surrounding odd numbers of marked points This isomorphism can be viewed as a finite presentation for Sp2g(Z)[2] since PB2g+1 is finitely presented and 2g+1 has a finite normal generating set. They have already been used by the last two authors to give finite presentations for certain congruence subgroups of SLn(Z) which are reminiscent of the standard presentation for SLn(Z); see [29]
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