The arc complex is not quasi-isometric to the sphere complex

Abstract

We show that the arc complex $\mathcal{A}(S_{g,1})$ is not quasi-isometric to the sphere complex ${\mathcal{S}}_{2g}$ associated to the double of a genus $2g$ handlebody. Along the way, we present a simple proof that $\mathcal{A}(S_{g,1})$ is quasi-isometrically rigid.

8 pages