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...) ? 



Quasi-isometric embeddings of the mapping class group into the Teichmuller space
A result of [Behrstock and Minsky](http://annals.math.princeton.edu/2008/167-3/p09) (cf. Hamenstadt too) implies that the rank of mapping class groups is the maximal rank of abelian subgroups, which is $3g+p-3$ for a connected hyperbolic surface of genus $g$ with $p$ boundary components. The rank of Teichmuller space with respect to the Weil-Petersson metric they show is $\lfloor \frac{3g+p-2}{2}\rfloor < 3g-3+p$, so there is no q.i. embedding of the mapping class group to Teichmuller space with the Weil-Petersson metric. 


On the other hand, [Eskin-Masur-Rafi](http://arxiv-web3.library.cornell.edu/abs/1307.3733) show that the rank of the Teichmuller metric is equal to that of the mapping class group. They point out that the orbit of a point under the mapping class group is not a quasi-isometric embedding. But this leaves open whether there might be a non-equivariant q.i. embedding. 


