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.


1 Answer 1


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.

  • $\begingroup$ Could you say a bit as to why their paper shows that? Skimming the paper they only specifically mention cyclic subgroups are undistorted. $\endgroup$
    – user35370
    Sep 27, 2018 at 16:11
  • $\begingroup$ 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. $\endgroup$
    – user35370
    Sep 27, 2018 at 16:43
  • $\begingroup$ @PaulPlummer: That is correct. $\endgroup$ Sep 27, 2018 at 19:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.