The curve graph, $ {\script G} $, associated to a compact surface $ \Sigma $ is the 1-skeleton of the curve complex defined by Harvey. Masur and Minsky showed that this graph is hyperbolic and defined the notion of a tight geodesic therein. We prove some finiteness results for such geodesics. For example, we show that a slice of the union of tight geodesics between any pair of points has cardinality bounded purely in terms of the topological type of $ \Sigma $. We deduce some consequences for the action of the mapping class group on $ {\script G} $. In paricular, we show that it satisfies an acylindricity condition, and that the stable lengths of pseudoanosov elements are rational with bounded denominator.
Invent. Math. 171 (2008) 281-300.