Combinatorics of tight geodesics and stable lengths

Webb, Richard

doi:10.1090/tran/6301

Combinatorics of tight geodesics and stable lengths
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by Richard C. H. Webb PDF

Trans. Amer. Math. Soc. 367 (2015), 7323-7342 Request permission

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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