Uniform hyperbolicity of the graphs of curves

Abstract

Let C.Sg;p/ denote the curve complex of the closed orientable surface of genus g with p punctures. Masur and Minksy and subsequently Bowditch showed that C.Sg;p/ is i–hyperbolic for some i D i.g;p/ . In this paper, we show that there exists some i > 0 independent of g;p such that the curve graph C1.Sg;p/ is i– hyperbolic. Furthermore, we use the main tool in the proof of this theorem to show uniform boundedness of two other quantities which a priori grow with g and p : the curve complex distance between two vertex cycles of the same train track, and the Lipschitz constants of the map from Teichmuller space to C.S/ sending a Riemann surface to the curve(s) of shortest extremal length.