Combinatorics of tight geodesics and stable lengths

We give an algorithm to compute the stable lengths of pseudoAnosovs 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 < a1 < a2 such that the minimal upper bound on ‘slices’ of tight geodesics is bounded below and above by a ξ(S) 1 and a ξ(S) 2 , where ξ(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.