Relative homology group of a translation surface

Question

Let $X$ be a Riemann surface of genus $g\ge 2$, $\omega$ an holomorphic differential on $X$ and $\Sigma(\omega)$ the set of zeroes of $\omega$.

We can equip $X$ with the singular flat metric $\omega\overline{\omega}$: the couple $(X,\omega\overline{\omega})$ is often called a translation surface.

Consider the relative homology group $H_1(X,\Sigma(\omega),\mathbb{C})$. Let $\gamma$ be a saddle connection on $X$: by definition it is a geodesic segment for $\omega\overline{\omega}$ which encounters the points of $\Sigma(\omega)$ only at its initial and terminal points. Consider $[\gamma]\in H_1(X,\Sigma(\omega),\mathbb{C})$ (the class of $\gamma$ in the relative homology group); I know that, since the metric $\omega\overline{\omega}$ is negatively curved, $\gamma$ is the only geodesic representative of $[\gamma]$.

My question is: what is the geodesic representative of $k[\gamma]$ for $k\in \mathbb{R}$?

Answer 1

I know that, since the metric $\omega\overline{\omega}$ is negatively curved, $\gamma$ is the only geodesic representative of $[\gamma]$.

is false (since you are working with relative homology, not the fundamental groupoid). Furthermore, asking for a geodesic representative of a real homology class is meaningless.