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Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at \(t=-1\). (English) Zbl 1328.57021

Let \(\Sigma_g\) be a closed oriented surface of genus \(g\). Its mapping class group \(\mathrm{Mod}_g\) acts on \(H_1(\Sigma_g;\mathbb{Z})\) and the kernel of its action is the Torelli group \(\mathcal{I}_g\). If we restrict ourselves to homeomorphisms which commute with a hyperelliptic involution \(\iota\), then we obtain the hyperelliptic mapping class group \(\mathrm{SMod}_g\); the hyperelliptic Torelli group \(\mathcal{SI}_g\) is, by definition, \(\mathrm{SMod}_g\cap \mathcal{I}_g\). The main result of the paper is Theorem A which states that for \(g\geq 0\) the group \(\mathcal{SI}_g\) is generated by Dehn twists about symmetric (with respect to \(\iota\)) separating curves. The authors note that Theorem A was known earlier for \(n\leq 6\).
The main idea of the proof of Theorem A is to reduce the problem to an investigation of braid groups and their representations. If we remove an \(\iota\)-invariant disk from \(\Sigma_g\) we obtain a surface \(\Sigma_g^1\) with a boundary. The involution \(\iota\) on \(\Sigma_g\) induces an involution \(\iota^1\) on \(\Sigma_g^1\) which fixes \(2g+1\) points. The images of the points under the factor-map \(\Sigma_g^1\to\Sigma_g^1/\iota^1\) allow us to consider \(\Sigma_g^1/\iota^1\) as a disk \(D_{2g+1}\) with \(2g+1\) marked points and to use the tool of representaions of the braid group \(B_{2g+1}\). In particular, the Burau representation \(\beta_{2g+1}\) of \(B_{2g+1}\) to \(\mathrm{GL}_{2g+1}(\mathbb{Z}[t,t^{-1}])\) is used for \(t=-1\). The representation \(\beta_n\) can be defined not only for odd \(n=2g+1\) but for all integers \(n\geq1\). Theorem C states that the group \(\mathcal{BI}_n\) which is the kernel of \(\beta_n: B_n\to\mathrm{GL}_{n}(\mathbb{Z})\) is generated by squares of Dehn twists about curves in \(D_n\) surrounding odd numbers of marked points. Theorem C implies Theorem A.
In turn, Theorem A gives the opportunity to describe generators of the fundamental group of the branched locus \(\widetilde{\mathcal{H}}_g\) of the period mapping from Torelli space to the Siegel upper half-plane. At last, as R. Hain showed [Proc. Symp. Pure Math. 74, 57–70 (2006; Zbl 1222.14014)], Theorem A immediately implies Theorem B: for \(g\geq 0\), each component of \(\widetilde{\mathcal{H}}_g^c\) is simply connected. Here \(\widetilde{\mathcal{H}}_g^c\) is obtained from \(\widetilde{\mathcal{H}}_g\) by adding hyperelliptic curves of compact type.

MSC:

57M60 Group actions on manifolds and cell complexes in low dimensions
57M05 Fundamental group, presentations, free differential calculus
57M12 Low-dimensional topology of special (e.g., branched) coverings
57M50 General geometric structures on low-dimensional manifolds

Citations:

Zbl 1222.14014
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References:

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