On the coarse geometry of the complex of domains

TL;DR

This paper investigates the coarse geometric properties of the complex of domains associated with a surface, revealing quasi-isometric relationships among various subcomplexes and providing new insights into their metric structures.

Contribution

It establishes isometric and quasi-isometric embeddings between the curve complex, arc complex, and domain complex, and offers new proofs of known quasi-isometry results.

Findings

The inclusion of the curve complex into certain subcomplexes is an isometric embedding.

The arc complex is quasi-isometric to a subcomplex of the domain complex for genus 0 surfaces.

The arc and curve complex is quasi-isometric to the curve complex.

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 paper 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 prove that the simplicial inclusion $P_\partial(S) \to D(S)$ is a quasi-isometric embedding if and only if $S$ has genus 0. 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 $A(S) \to AC(S)$.