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A translation surface is a Riemann surface equipped with a holomorphic 1-form $\omega$ and a Riemannian metric $g=\omega \bar \omega$ with conical singularities. It is well-known that there exists closed regular geodesics, i.e., those not going through singular points, but usually shortest curves in a given homotopy class pass through the singular points.

Question 1: Is the first homology group generated by regular closed geodesics (at least over $\mathbb Q$)? I am mainly interested in the case of genus $5$ surfaces with $\omega$ having four distinct double zeros.

Question 2: If question 1 has a negative answer, I would like to know if there is another way to determine the periods of $\omega$ in terms of the lengths of the regular closed geodesics $\gamma$ and their angles (defined by the (constant) modulus of $\omega(\gamma')$).

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