Effective geometry of curve graphs

We show that curve graphs and arc graphs are uniformly hyperbolic i.e. all of these graphs are hyperbolic with some fixed hyperbolicity constant. This is joint work with Sebastian Hensel and Piotr Przytycki. Note that curve graphs were shown to be hyperbolic by Masur and Minsky, arc graphs were shown to be hyperbolic by Masur and Schleimer, and curve graphs were shown to be uniformly hyperbolic by Aougab, Bowditch and Clay-Rafi-Schleimer independently.

We show that if every vertex of a fixed geodesic in the curve graph cuts some fixed subsurface then the image of that geodesic under the corresponding subsurface projection has its diameter bounded from above by 62 (or 50) if the subsurface is an annulus (or otherwise). This is an effective version of the bounded geodesic image theorem of Masur and Minsky, indeed, there is a universal constant that can be taken as a bound.

In terms of the complexity of a surface we provide an exponential upper bound on the so-called slices of tight geodesics. This is an effective version of a theorem of Bowditch. Using this we provide algorithms to compute tight geodesic axes of pseudo-Anosovs in the curve graph.

All of our proofs are combinatorial in nature. In particular, we do not use any geometric limiting arguments.