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Quasi-isometric embeddings of the mapping class group into the Teichmuller space
Does there exist a quasi-isometric embedding
$$MCG(S) \to (\mathrm{Teich}(S), d)$$
for $d$ any "known" distance on the Teichmuller space (i.e. Teichmuller, Weil-Petersson, Thurston...) ?
