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Subgroups generated by two pseudo-Anosov elements in a mapping class group. II. Uniform bound on exponents


Author: Koji Fujiwara
Journal: Trans. Amer. Math. Soc. 367 (2015), 4377-4405
MSC (2010): Primary 20F65; Secondary 20F67, 20F38, 20F28
Published electronically: December 24, 2014
MathSciNet review: 3324932
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Abstract: Let $ S$ be a compact orientable surface, and $ \mathrm {Mod}(S)$ its mapping class group. Then there exists a constant $ M(S)$, which depends on $ S$, with the following property. Suppose $ a,b \in \mathrm {Mod}(S)$ are independent (i.e., $ [a^n,b^m]\not =1$ for any $ n,m \not =0$) pseudo-Anosov elements. Then for any $ n,m \ge M$, the subgroup $ \langle a^n,b^m \rangle $ is a free group freely generated by $ a^n$ and $ b^m$, and convex-cocompact in the sense of Farb-Mosher. In particular all non-trivial elements in $ \langle a^n,b^m \rangle $ are pseudo-Anosov. We also show that there exists a constant $ N$, which depends on $ a,b$, such that $ \langle a^n,b^m \rangle $ is a free group freely generated by $ a^n$ and $ b^m$, and convex-cocompact if $ \vert n\vert+\vert m\vert \ge N$ and $ nm \not =0$.


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Additional Information

Koji Fujiwara
Affiliation: Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan
Email: kfujiwara@math.kyoto-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-2014-06292-2
Received by editor(s): September 13, 2009
Received by editor(s) in revised form: May 1, 2012, September 5, 2013, and September 28, 2013
Published electronically: December 24, 2014
Dedicated: In memory of Professor Shoshichi Kobayashi
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.