Let $S\_{g,b}$ be an oriented surface with $b$ boundary components and $S\_g^b$ be an oriented surface with $b$ punctures. Let $\mathrm{Mod}(S\_{g,b})$ and $\mathrm{Mod}(S\_g^b)$ their (orientation preserving) mapping class group fixing boundary/punctures pointwise. There is a short exact sequence: 


$$1 \to \mathbb Z^b \to \mathrm{Mod}(S\_{g,b}) \to \mathrm{Mod}(S\_g^b) \to 1$$


1. Is $\mathbb Z^b$ undistorted in $\mathrm{Mod}(S\_{g,b})$ ?
2. What is the maximal rank of an undistorted Abelian subgroup in $\mathrm{Mod}(S\_g^b)$ ?


Thank you in advance. 



Distorsion of subgroups of the mapping class group
Farb, Lubotzky, and Minsky proved in


B. Farb, A. Lubotzky and Y. Minsky, Rank-1 phenomena for mapping class groups, Duke
Math. J. 106 (2001), no. 3, 581–597.


that all abelian subgroups of the mapping class group are undistorted.


