Strong contractability of geodesics in the mapping class group

A geodesic is strongly contracting if its nearest point projection takes disjoint balls from the geodesic to sets of bounded diameter, where the bound is independent of the ball. In joint work with Kasra Rafi, we show that the axis of a pseudo-Anosov homeomorphism in the mapping class group may not have the strong contractibility property. In particular, we show that it is possible to choose an appropriate generating set for the mapping class group of the five-times punctured sphere so that there exists a pseudo-Anosov homeomorphism $\phi$, a sequence of points $x_k$, and a sequence of radii $R_k$ so that the ball $B_{R_k} (x_k)$ is disjoint from the axis of $\phi$, but the closest point projection of $B_{R_k} (x_k)$ to the axis is at least $c \log(R_k)$. Along the way, we show that it is, in fact, possible to construct explicit geodesics in the mapping class group.

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