Combinatorics of tight geodesics and stable lengths

TL;DR

This paper introduces algorithms for computing stable lengths and invariant axes of pseudo-Anosovs on the curve graph, providing new bounds and insights into the geometry of surface mapping class groups.

Contribution

It presents the first algorithms for stable length computation and invariant axes, along with bounds on tight geodesic slices and asymptotic dimensions.

Findings

Algorithms for stable length computation of pseudo-Anosovs

Bounds on slices of tight geodesics based on surface complexity

First computable bounds on asymptotic dimension of curve graphs

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.