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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 Abelian 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 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.

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Anonymous
  • 828
  • 6
  • 14

Distorsion of subgroups of the mapping class group

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.