Dominating CAT-1 surface group representations by Fuchsian ones.

We show that for every representation ρ : π 1 (S g) → Isom (X) of the fundamental group of a genus g ≥ 2 surface to the isometry group of a complete CAT - 1 metric space X there exists a Fuchsian representation j and a (j , ρ) -equivariant map from H 2 to X which is c -Lipschitz for some c < 1 , or ρ restricts to a Fuchsian representation. [ABSTRACT FROM AUTHOR]