Given a finitely generated subgroup of a finitely generated hyperbolic group. Is it true that the inclusion of each subgroup is a quasiisometric embedding ?


The first example for a group that does not have this property is a Baumslag-Solitar group $BS(1,m)= \langle a,b| bab^{-1}=a^m\rangle$. We have $a^{m^k}=b^kab^{-k}$ for each $k$. This shows for example that the inclusion of the subgroup generated by $a$ is not a quasiisometric embedding.


Then one can consider the class of groups with the property that the inclusion of any subgroup is a quasiisometric embedding. Has this class been studied? 



Are subgroups of hyperbolic groups quasiisometrically embedded ?
The answer to the first question is "no". Look at the Rips' construction <http://en.wikipedia.org/wiki/Small_cancellation_theory>. The finitely generated normal subgroup there is usually very badly distorted. I do not see many general properties of the class of groups where every f.g. subgroup is undistorted. It contains so diverse groups as free groups and Tarski monsters. Studying hyperbolic groups with this property makes more sense but I doubt anything general is known about this class either. 


