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