Super-maximal representations from fundamental groups of punctured surfaces to $\mathrm{PSL}(2,\mathbb{R})$

We study a particular class of representations from the fundamental groups of punctured spheres $Σ_{0,n}$ to the group $\text{PSL} (2,\mathbb R)$ (and their moduli spaces), that we call \emph{super-maximal}. Super-maximal representations are shown to be \emph{totally non hyperbolic}, in the sense that every simple closed curve is mapped to a non hyperbolic element. They are also shown to be \emph{geometrizable} (appart from the reducible super-maximal ones) in the following very strong sense : for any element of the Teichmüller space $\mathcal T_{0,n}$, there is a unique holomorphic equivariant map with values in the lower half-plane $\mathbb H^-$. In the relative character variety, the components of super-maximal representations are shown to be compact, and symplectomorphic (with respect to the Atiyah-Bott-Goldman symplectic structure) to the complex projective space of dimension $n-3$ equipped with a certain multiple of the Fubiny-Study form that we compute explicitly (this generalizes a result of Benedetto--Goldman for the sphere minus four points). Those are the unique compact components in relative character varieties of $\text{PSL}(2,\mathbb R)$. This latter fact will be proved in a companion paper.

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