Fuch曲面群表示对CAT（-1）曲面群表示的控制
Dominating CAT(-1) surface group representations by Fuchsian ones
摘要：
证明了对于亏格$ g \ge 2 $曲面的基本群到完备$ \text{CAT}(-1) $度量空间$ X $的等距群的每一个表示$ ρ: π_{1} (S_{g}) \to \text{Isom}(X) $，存在Fuchsian表示$ j $和从$ \mathbb{H}^{2} $到$ X $的$ (j, ρ) $ -等变映射，对于某些$ c $ -Lipschitz，或$ ρ$限制于Fuchsian表示.这推广了Gueritaud-Kassel-Wolff，Deroin-Tholozan和Daskalopoulos-Mese-Sanders-Vdodorf的结果
英文摘要：
We show that for every representation $ ρ: π_{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, ρ) $-equivariant map from $ \mathbb{H}^{2} $ 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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