On the coarse geometry of the complex of domains

  • Valentina Disarlo

    Université de Strasbourg, France
On the coarse geometry of the complex of domains cover

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The complex of domains D(S)D(S) is a geometric tool with a very rich simplicial structure, it contains the curve complex C(S)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 Δ(S)\Delta(S) of D(S)D(S) which contains the curve complex C(S)C(S), the natural simplicial inclusion C(S)Δ(S)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)A(S) is quasi-isometric to the subcomplex P(S)P_\partial(S) of D(S)D(S) spanned by the vertices which are peripheral pair of pants, and we give necessary and sufficient conditions on SS for the simplicial inclusion P(S)D(S)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)AC(S). We give a new proof of the fact that AC(S)AC(S) is quasi-isometric to C(S)C(S), and we discuss the metric properties of the simplicial inclusion A(S)AC(S)A(S) \to AC(S).