## A question about lifting of quasiconformal map to H

Let $f, g :X \to Y$ be two homotopic quasiconformal homeomorphism of ( not necessarily closed ) hyperbolic Riemann surfaces. How to prove : there exists lifts $\tilde{f},\tilde{g}$ of $f,g$ respectively such that those two lifts agree on the real line $R=$ boundary of $H$. [It is a homework from a book]. I guess it is enough to assume $X=Y, g=Id_X,$. and the fact that we can extend the maps to from $H$ to$\bar{H}$ follows from Holder continuity ( implying uniform continuity ) of a q.c. map on $H$.Am I right ?

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