Mapping class group dynamics and the holonomy of branched affine structures

We classify, up to few exceptions, the orbit closures of the $\mathrm{Mod}(Σ)$-action on the affine character variety $χ(\mathrm{Aff}(\mathbb{C}))$. We obtain from this classification that the only obstruction for a non-abelian representation $ρ: π_1 Σ\longrightarrow \mathrm{Aff}(\mathbb{C})$ to be the holonomy of a branched affine structure on $Σ$ is to be Euclidean and not to have positive volume, where $Σ$ is a closed oriented surface of genus $g \geq 2$.

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