Geodesics in the mapping class group

TL;DR

This paper constructs explicit geodesics in the mapping class group, demonstrating that their shadows in the curve graph are not necessarily quasi-geodesics and that pseudo-Anosov axes may lack strong contractibility.

Contribution

It provides explicit examples showing limitations of geometric properties in the mapping class group and its actions on associated complexes.

Findings

Shadows of geodesics in the curve graph are not always quasi-geodesics.

Pseudo-Anosov axes may lack strong contractibility.

Projection diameters can grow logarithmically with radius.

Abstract

We construct explicit examples of geodesics in the mapping class group and show that the shadow of a geodesic in mapping class group to the curve graph does not have to be a quasi-geodesic. We also show that the quasi-axis of a pseudo-Anosov element of the mapping class group may not have the strong contractibility property. Specifically, we show that, after choosing a generating set carefully, one can find a pseudo-Anosov homeomorphism f, a sequence of points w_k and a sequence of radii r_k so that the ball B(w_k, r_k) is disjoint from a quasi-axis a of f, but for any projection map from mapping class group to a, the diameter of the image of B(w_k, r_k) grows like log(r_k).