(2002) Leasure, Jason Paige; Gordon, Cameron, 1945-

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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.