Geodesic
On the topology of $\mathcal{H}(2)$
Duc-Manh Nguyen1
1Université de Bordeaux I, Talence, France
Groups, geometry, and dynamics, Tome 8 (2014) no. 2, pp. 513-551

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DOI

The space H(2) consists of pairs (M,ω), where M is a Riemann surface of genus two, and ω is a holomorphic 1-form which has only one zero of order two. There exists a natural action of C∗ on H(2) by multiplication to the holomorphic 1-form. In this paper, we single out a proper subgroup Γ of Sp(4,Z) generated by three elements, and show that the space H(2)/C∗ can be identified with the quotient Γ\J2​, where J2​ is the Jacobian locus in the Siegel upper half space H2​. A direct consequence of this result is that [Sp(4,Z):Γ]=6. The group Γ can also be interpreted as the image of the fundamental group of H(2)/C∗ in the symplectic group Sp(4,Z).
DOI : 10.4171/ggd/237
Classification : 57-XX
Mots-clés : Riemann surface, translation surface

Duc-Manh Nguyen  1

1 Université de Bordeaux I, Talence, France 
Duc-Manh Nguyen. On the topology of $\mathcal{H}(2)$. Groups, geometry, and dynamics, Tome 8 (2014) no. 2, pp. 513-551. doi: 10.4171/ggd/237
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     year = {2014},
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     number = {2},
     doi = {10.4171/ggd/237},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/237/}
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