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Let $S$ be an orientable surface with negative Euler characteristic. Can somebody provide a reference for the following well-known results:

  1. the cyclic subgroup generated by a pseudo-Anosov element in the mapping class $MCG(S)$ is undistorted in the mapping class group.

  2. the cyclic subgroup generated by a Dehn twist of the mapping class group is undistorted.

  3. $\pi_1(S)$ is exponentially distorted in the mapping class group.

Thank you in advance.

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up vote 6 down vote accepted

The following paper proves that all infinite cyclic subgroups of the mapping class group are undistorted (subsuming questions 1 and 2):

Farb, Benson; Lubotzky, Alexander; Minsky, Yair, Rank-1 phenomena for mapping class groups. Duke Math. J. 106 (2001), no. 3, 581–597.

For your third question, you don't specify which surface group you are talking about. I'm going to assume that you are asking about the "point-pushing subgroup", in which case the result you are looking for is in the following paper:

Broaddus, Nathan; Farb, Benson; Putman, Andrew, Irreducible Sp-representations and subgroup distortion in the mapping class group. Comment. Math. Helv. 86 (2011), no. 3, 537–556.

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