Mapping class group dynamics and the holonomy of branched affine structures

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 $\rho :{\pi }_1\Sigma ⟶\mathrm{Aff}(\mathbb{C})$ 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 $g\ge 2$.

26 pages, 4 figures