COMBINATORICS OF TIGHT GEODESICS AND STABLE LENGTHS

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 < alpha 1 < alpha 2 such that the minimal upper bound on 'slices' of tight geodesics is bounded below and above by alpha(xi( S))(1) and alpha(xi(S))(2) ,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.