Density results for the modular group of infinite-type surfaces

In this work we show two results about approximating, with respect to the compact-open topology, mapping classes on surfaces of infinite-type by quasi-conformal maps, in particular we are interested in density results. The first result is that given any infinite-type surface S there exists a hyperbolic structure X on S such that PMCG(S)⊆ Mod(X)̄, for Mod(X) the set of quasi-conformal homeomorphism on X. The second result is that given any surface S with countably many ends then there exists a hyperbolic structure X such that MCG (S)=Mod(X)̄.