Combinatorics of tight geodesics and stable lengths
Richard Webb
Source 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 1>a_1>a_2 such that the minimal upper bound on ‘slices’ of tight geodesics is bounded below and above by a 1 ξ ( S ) a_1^{\xi (S)} and a 2 ξ ( S ) a_2^{\xi (S)} , where ξ ( S ) \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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