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

TL;DR

This paper demonstrates that the arc complex of a surface with boundary is not quasi-isometric to the sphere complex of a related handlebody, and also proves the quasi-isometric rigidity of the arc complex.

Contribution

It establishes the non-quasi-isometry between the arc complex and the sphere complex and provides a simple proof of the arc complex's quasi-isometric rigidity.

Findings

Arc complex is not quasi-isometric to the sphere complex.

Arc complex is quasi-isometrically rigid.

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.