| Description |
Let Sg,n be a compact surface of g genus and n boundary components. Let x(Sg,n) = 3g + n − 3 be the complexity of the surface. Our main space in this dissertation is the curve complex C(S) by Harvey. The curve complex is known to be a Gromov hyperbolic, infinite diameter, and locally infinite space. Our main object in this dissertation is tight geodesics by Masur-Minsky. The curve complex plays an important role in the study of low dimensional topology and geometry. Especially, the Masur-Minsky theory of hierarchies of the curve complex gave a complete understanding on the large-scale geometry of the mapping class groups with other important tools such as tight geodesics and subsurface projections. Bowditch studied the cardinality of slices of tight geodesics and as its applications, he showed that the mapping class groups act on the curve complex acylindrically and that the stable lengths of pseudo-anosov elements are rational with bounded denominator. However, since his proof is done by a geometric limit argument via hyperbolic 3-manifolds, his argument does not give a computable bound for the cardinality of slices of tight geodesics. In this dissertation, we extend Bowditch's result. In Chapter 2, we show there exists a computable bound of slices of tight geodesics which only depends on the surface and the distance between initial and terminal curves by a combinatorial approach. In Chapter 3, we show there exists a computable bound of slices of tight geodesics which only depends on the surface. Indeed, the second statement is a direct corollary of the following new result. While the curve complex is locally infinite, we show that it is "locally finite" under subsurface projections. We define the local finiteness property; this is a property which any locally finite graph whose diameter is infinite with a uniformly bounded valency satisfies. Suppose X is a such graph. Let dX be the simplicial metric on X. Local finiteness property: Given l > 0 and k > 1, there exists a computable N(l, k, valency of X) > 0 such that for any C ✓ X, if |C| > N, then there exist {xj} ⇢ C such that |{xj}| % k so that dX(xs, xt) > l for all s, t such that s 6= t. The local finiteness property does not hold for the curve complex since it is locally infinite. However, by using subsurface projections, we show Local finiteness property of the curve complex via subsurface projections: Given l > 0 and k > 1, there exists a computable N(l, k, x(S)) > 0 such that for any C ✓ C(S), if |C| > N, then there exists {xj} ✓ C such that |{xj}| % k and Z ✓ S so that dZ(xs, xt) > l for all s, t such that s 6= t. As a corollary of the above main result with a special behavior of tight geodesics, we give a computable bound of slices of tight geodesics which only depends on the surface. Lastly, we define a new class of geodesics, weak tight geodesics. Indeed, we can use the local finiteness property of the curve complex via subsurface projections to show that the cardinality of slices of weak tight geodesics are bounded by W(Wx(S)) for some constant W which only depends on the surface. |
