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

Abstract

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 the isometry group of a complete $ \text{CAT}(-1) $ metric space $ X $ there exists a 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 of Gueritaud-Kassel-Wolff, Deroin-Tholozan and Daskalopoulos-Mese-Sanders-Vdovina