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

2020/11/08 by Florestan Martin-Baillon, Martin-Baillon, Florestan
Mathematics · #Geometric and Algebraic Topology #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2011.03954

Abstract

We show that for every representation ρ: π₁ (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 ℍ² to X which is c -Lipschitz for some c < 1 , or ρ restricts to a Fuchsian representation. This generalizes results of Gueritaud-Kassel-Wolff, Deroin-Tholozan and Daskalopoulos-Mese-Sanders-Vdovina

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