#### Volume 21, issue 6 (2021)

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Geodesics in the mapping class group

### Kasra Rafi and Yvon Verberne

Algebraic & Geometric Topology 21 (2021) 2995–3017
##### Abstract

We construct explicit examples of geodesics in the mapping class group and show that the shadow of a geodesic in the mapping class group to the curve graph does not have to be a uniform quality quasigeodesic. We also show that the quasiaxis of a pseudo-Anosov element of the mapping class group may not have the strongly contracting property. Specifically, we show that, after choosing a generating set carefully, one can find a pseudo-Anosov homeomorphism $\varphi$, a sequence of points ${w}_{k}$ and a sequence of radii ${r}_{k}$ such that the ball $B\left({w}_{k},{r}_{k}\right)$ is disjoint from a quasiaxis  ${\mathit{a}}_{\varphi }$ of $\varphi$, but, for any projection from the mapping class group to ${\mathit{a}}_{\varphi }$, the diameter of the image of $B\left({w}_{k},{r}_{k}\right)$ grows like $log\left({r}_{k}\right)$.

##### Keywords
homeomorphisms of surfaces, mapping class group, pseudo-Anosov homeomorphisms, geodesic, contracting
##### Mathematical Subject Classification 2010
Primary: 20F34, 37E30, 57M07