Abstract
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 ge 2.
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