= Hyperbolic holonomy representation
{title2=$\rho:\pi_1(\Sigma_g)\to\operatorname{PSL}_2(\mathbb R)$}
For a marked closed <hyperbolic surface>, the representation of its <fundamental group> whose image acts as the deck group on the <hyperbolic plane>. It is discrete and faithful, determined up to conjugation by the marking and the metric, and gives $X=\mathbb H^2/\rho(\pi_1\Sigma_g)$. For a closed orientable surface it admits determinant-one lifts. It is different from the tangent-space action of the <Riemannian holonomy group>.
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