# Geodesics in the complex of curves of a surface

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## Abstract

The curve complex of a closed surface S of genus g ≥ 2, C(S), is the complex whose vertices are isotopy classes of simple closed curves on S, and ([x0], . . . , [xn]) is a simplex of C if and only if there are disjoint representatives xi and xj for all i, j. The curve complex of the torus is similar, with ([x0], . . . , [xn]) a simplex if and only if there are representatives xi and xj which meet in a single point for each i = j. We use the path metric on C(S). This dissertation introduces several tools for studying geodesics in the curve complexes of closed orientable surfaces. In the simplest case, when S is a torus, we prove a structure theorem for C(S) and deduce some results about its global geometry. For the higher genus cases, we introduce two methods for approximating distances. The first yields an elementary proof of the known result [10], [6] that the curve complex has infinite diameter, and a constructive means for estimating distance. The second bounds certain intersection numbers and results in an algorithm to compute distance precisely. All results are expressly constructive and elementary.