On the coarse geometry of the complex of domains

Abstract

The complex of domains $D(S)$ is a geometric tool with a very rich simplicial structure, it contains the curve complex $C(S)$ as a simplicial subcomplex. In this chapter we shall regard it as a metric space, endowed with the metric which makes each simplex Euclidean with edges of length 1, and we shall discuss its coarse geometry. We prove that for every subcomplex $\Delta (S)$ of $D(S)$ which contains the curve complex $C(S)$, the natural simplicial inclusion $C(S)\to \Delta (S)$ is an isometric embedding and a quasi-isometry. We prove that, except a few cases, the arc complex $A(S)$ is quasi-isometric to the subcomplex $P_{\partial }(S)$ of $D(S)$ spanned by the vertices which are peripheral pair of pants, and we give necessary and sufficient conditions on $S$ for the simplicial inclusion $P_{\partial }(S)\to D(S)$ to be a quasi-isometric embedding. We then apply these results to the arc and curve complex $AC(S)$. We give a new proof of the fact that $AC(S)$ is quasi-isometric to $C(S)$, and we discuss the metric properties of the simplicial inclusion $A(S)\to AC(S)$.