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We show that for every representation $ \rho : \pi_{1} (S_{g}) \to\text{Isom}(X) $ of the fundamental group of a genus $ g \ge 2 $ surface to theisometry group of a complete $ \text{CAT}(-1) $ metric space $ X $ there existsa Fuchsian representation $ j $ and a $ (j, \rho) $-equivariant map from $\mathbb{H}^{2} $ to $ X $ which is $ c $ -Lipschitz for some $ c < 1 $, or $\rho $ restricts to a Fuchsian representation. This generalizes results ofGueritaud-Kassel-Wolff, Deroin-Tholozan and Daskalopoulos-Mese-Sanders-Vdovina

Dominating CAT(-1) surface group representations by Fuchsian ones