We discuss about flat structures on surfaces of finite type $S_{g,n}$, possibly with punctures. For a given representation $\chi\colon \pi_1(S_{g,n})\to \textnormal{Aff}(\mathbb C)$, we wonder if there exists a flat structure having the given representations as the holonomy representation. For closed surfaces $(n=0)$, holonomy representations has been determined by works of Haupt for representations in $\mathbb C$ and subsequently by Ghazouani for a generic representation in $\textnormal{Aff}(\mathbb C)$. It turns out that for surfaces of hyperbolic type, i.e. $2-2g-n<0$, the resulting structures must have special points, called branched points, around which the geometry fails to be modelled on $\mathbb C$. In the present seminar we discuss the case of punctured surfaces and provide conditions under which a representation $\chi$ is a holonomy representation of some flat structure. In this case, being surfaces no longer closed, it is even possible to find flat structures with no branched points. This is a joint work with Subhojoy Gupta and partially with Shabarish Chenakkod.

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Last updated: 12 Apr 2024