On the coarse geometry of the complex of domains
Valentina DisarloUniversité de Strasbourg, France
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The complex of domains is a geometric tool with a very rich simplicial structure, it contains the curve complex 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 of which contains the curve complex , the natural simplicial inclusion is an isometric embedding and a quasi-isometry. We prove that, except a few cases, the arc complex is quasi-isometric to the subcomplex of spanned by the vertices which are peripheral pair of pants, and we give necessary and sufficient conditions on for the simplicial inclusion to be a quasi-isometric embedding. We then apply these results to the arc and curve complex . We give a new proof of the fact that is quasi-isometric to , and we discuss the metric properties of the simplicial inclusion .