Given a closed orientable surface $S$, a $(G,X)-$structure on $S$ is the datum of a maximal atlas whose charts take values on $X$ and transition functions are restrictions of elements in $G$. Any such structure induces a holonomy representation $\rho:\pi_1(\widetilde{S})\to X$ which encodes geometric data of the structure. Conversely, can we recover a geometric structure from a given representation? Is such a structure unique? In this talk we answer these questions by providing old and new results.

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Last updated: 06 Mar 2020