Groups, Geometry, and Dynamics

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Volume 8, Issue 2, 2014, pp. 513–551
DOI: 10.4171/GGD/237

Published online: 2014-07-08

On the topology of $\mathcal{H}(2)$

Duc-Manh Nguyen[1]

(1) Université de Bordeaux I, Talence, France

The space $\mathcal{H}(2)$ consists of pairs $(M,\omega)$, where $M$ is a Riemann surface of genus two, and $\omega$ is a holomorphic 1-form which has only one zero of order two. There exists a natural action of $\mathbb{C}^*$ on $\mathcal{H}(2)$ by multiplication to the holomorphic 1-form. In this paper, we single out a proper subgroup $\Gamma$ of $\mathrm{Sp}(4,\mathbb{Z})$ generated by three elements, and show that the space $\mathcal{H}(2)/\mathbb{C}^*$ can be identified with the quotient $\Gamma\backslash\mathcal{J}_2$, where $\mathcal{J}_2$ is the Jacobian locus in the Siegel upper half space $\mathfrak{H}_2$. A direct consequence of this result is that $[\mathrm{Sp}(4,\mathbb{Z}):\Gamma]=6$. The group $\Gamma$ can also be interpreted as the image of the fundamental group of $\mathcal{H}(2)/\mathbb{C}^*$ in the symplectic group $\mathrm{Sp}(4,\mathbb{Z})$.

Keywords: Riemann surface, translation surface

Nguyen Duc-Manh: On the topology of $\mathcal{H}(2)$. Groups Geom. Dyn. 8 (2014), 513-551. doi: 10.4171/GGD/237