We study the action of the mapping class group of a surface on the 1-skeleton of Harvey's curve complex from a computational perspective. With the appropriate quantification, we find that the number of mapping classes moving a long geodesic path a small distance is explicitly bounded in terms of certain intersection numbers and the topological type of the surface.

The author wishes to thank the Centre Interfacultaire Bernoulli EPFL for its hospitality and to thank Brian Bowditch for many interesting conversations. This work was partially supported by the World Premier International Research Center Initiative, MEXT in Japan. The author also gratefully acknowledges the partial support by a JSPS Grant-in-Aid for Young Scientists. The author wishes to thank the referee for reading this paper and for making many useful suggestions.