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A lower bound of the distortion of the Torelli group in the mapping class group with boundary components

Erika Kuno

Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan

E-mail Address: kuno.e.aa@m.titech.ac.jp

Corresponding author.

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and

Genki Omori

Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan

E-mail Address: omori.g.aa@m.titech.ac.jp

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https://doi.org/10.1142/S0218216517500493Cited by:0 (Source: Crossref)

Abstract

We prove that the Torelli group of an oriented surface with any number of boundary components is at least exponentially distorted in the mapping class group by using Broaddus–Farb–Putman’s techniques. Further we show that the distortion of the Torelli group in the level $d$ $d$ mapping class group is the same as that in the mapping class group.

Keywords:

AMSC: 20F65, 20F38, 57M07

References

1. N. Broaddus, B. Farb and A. Putman, Irreducible Sp-representations and subgroup distortion in the mapping class group, Comment. Math. Helv. 86(3) (2011) 537–556. Crossref, Web of Science, Google Scholar
2. D. B. Cohen, The Dehn function of $Sp (2 n; ℤ)$ , J. Topol. Anal. 9(2) (2017) 225–290. Link, Web of Science, Google Scholar
3. U. Hamenstädt, Geometry of the mapping class group I: Boundary amenability, Invent. Math. 175(3) (2009) 545–609. Crossref, Web of Science, Google Scholar
4. U. Hamenstädt and S. Hensel, The geometry of the handlebody groups I: Distortion, J. Topol. Anal. 4(1) (2012) 71–97. Link, Web of Science, Google Scholar
5. H. Masur and Y. Minsky, Geometry of the complex of curves II: Hierarchical structure, Geom. Funct. Anal. 10(4) (2000) 902–974. Crossref, Web of Science, Google Scholar
6. A. Putman, Cutting and pasting in the Torelli group, Geom. Topol. 11 (2007) 829–865. Crossref, Web of Science, Google Scholar