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

Publication date: 8 November 2020

Abstract: We show that for every representation

h o : p i_{1} (S_{g}) o e x t I s o m (X)

of the fundamental group of a genus

g g e 2

surface to the isometry group of a complete

e x t C A T (- 1)

metric space

X

there exists a Fuchsian representation

j

and a

(j, h o)

-equivariant map from

m a t h b b H^{2}

to

X

which is

c

-Lipschitz for some

c < 1

, or

h o

restricts to a Fuchsian representation. This generalizes results of Gueritaud-Kassel-Wolff, Deroin-Tholozan and Daskalopoulos-Mese-Sanders-Vdovina

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