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

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.

• Could you say a bit as to why their paper shows that? Skimming the paper they only specifically mention cyclic subgroups are undistorted.
– user35370
Sep 27 '18 at 16:11
• I guess it is also because abelian subgroups, up to finite index, are generated by elements supported on disjoint subsurfaces. I think this is done in Abelian and solvable subgroups of the mapping class group by Birman, Lubotzky, McCarthy.
– user35370
Sep 27 '18 at 16:43
• @PaulPlummer: That is correct. Sep 27 '18 at 19:15