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 Abelian undistorted subgroup in $\mathrm{Mod}(S_g^b)$ ?

Thank you in advance.

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.