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

TL;DR

This paper proves that any surface group representation into a CAT(-1) space can be dominated by a Fuchsian representation via a Lipschitz equivariant map, extending previous results in geometric group theory.

Contribution

It generalizes prior work by establishing a Lipschitz domination of surface group representations by Fuchsian groups in CAT(-1) spaces.

Findings

Existence of a c-Lipschitz equivariant map for some c<1

Either the representation is dominated or it is Fuchsian

Extends previous domination results to broader CAT(-1) spaces

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