Mapping class group dynamics and the holonomy of branched affine structures

We classify, up to few exceptions, the orbit closures of the $\mathrm{Mod}(\Sigma)$-action on the affine character variety $\chi(\mathrm{Aff}(\mathbb{C}))$. We obtain from this classification that the only obstruction for a non-abelian representation $\rho : \pi_1 \Sigma \longrightarrow \mathrm{Aff}(\mathbb{C})$ to be the holonomy of a branched affine structure on $\Sigma$ is to be Euclidean and not to have positive volume, where $\Sigma$ is a closed oriented surface of genus $g \geq 2$.