# Harmonic equivariant maps and Simpson's correspondence

Let $\Gamma\subset PSL(2,R)$ be a Fuchsian group. For which representations $\rho:\Gamma\to PSL(2,R)$ does there exist a harmonic map from the hyperbolic plane to itself satisfying

$f(\gamma z)=\rho(\gamma)f(z)$

for all $\gamma\in \Gamma$?

I was told that when $\Gamma$ is co-compact and torsion free, the existence of such an $f$ was given by Simpson's correspondence. I am interested to know if this extends in any way to more general $\Gamma$, especially co-finte volume $\Gamma$, and if this fact can be deduced in an elementary way.

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For non-cocompact groups, I presume, you are interested in finite-energy harmonic maps. For lattices $\Gamma\subset PSL(2, {\mathbb R})$, see for instance the paper by Jost, Yang, Zuo: http://arxiv.org/pdf/math/0505144v1.pdf (appeared in J. Reine Angew. Math. 609 (2007), 137--159) where more general target spaces are discussed as well.