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Transactions of the American Mathematical Society

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Combinatorics of tight geodesics and stable lengths


Author: Richard C. H. Webb
Journal: Trans. Amer. Math. Soc. 367 (2015), 7323-7342
MSC (2010): Primary 57M99; Secondary 20F65
DOI: https://doi.org/10.1090/tran/6301
Published electronically: April 3, 2015
MathSciNet review: 3378831
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Abstract: We give an algorithm to compute the stable lengths of pseudo-Anosovs on the curve graph, answering a question of Bowditch. We also give a procedure to compute all invariant tight geodesic axes of pseudo-Anosovs.

Along the way we show that there are constants $ 1<a_1<a_2$ such that the minimal upper bound on `slices' of tight geodesics is bounded below and above by $ a_1^{\xi (S)}$ and $ a_2^{\xi (S)}$, where $ \xi (S)$ is the complexity of the surface. As a consequence, we give the first computable bounds on the asymptotic dimension of curve graphs and mapping class groups.

Our techniques involve a generalization of Masur-Minsky's tight geodesics and a new class of paths on which their tightening procedure works.


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Additional Information

Richard C. H. Webb
Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
Address at time of publication: Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom
Email: R.C.H.Webb@warwick.ac.uk, richard.webb@ucl.ac.uk

DOI: https://doi.org/10.1090/tran/6301
Received by editor(s): May 27, 2013
Received by editor(s) in revised form: October 13, 2013
Published electronically: April 3, 2015
Additional Notes: This work was supported by the Engineering and Physical Sciences Research Council Doctoral Training Grant.
Article copyright: © Copyright 2015 American Mathematical Society