Combinatorics of tight geodesics and stable lengths

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.

19 pages, 2 figures

References in corpus (6)

• Uniform hyperbolicity of the curve graph via surgery sequences
• Slim unicorns and uniform hyperbolicity for arc graphs and curve graphs
• Asymptotic Translation Length in the Curve Complex
• Uniform Hyperbolicity of the Graphs of Curves
• Invariant geodesics in the curve complex under point-pushing pseudo-Anosov mapping classes
• On the minimum of asymptotic translation lengths of point-pushing pseudo-Anosov maps on one punctured Riemann surfaces

Cited by in corpus (1)

• MICC: A tool for computing short distances in the curve complex