Uniform hyperbolicity of the graphs of curves

Abstract

Let $\mathcal{C}\left(S_{g,p}\right)$ denote the curve complex of the closed orientable surface of genus $g$ with $p$ punctures. Masur and Minksy and subsequently Bowditch showed that $\mathcal{C}\left(S_{g,p}\right)$ is $\delta$–hyperbolic for some $\delta =\delta \left(g,p\right)$. In this paper, we show that there exists some $\delta >0$ independent of $g,p$ such that the curve graph ${\mathcal{C}}_1\left(S_{g,p}\right)$ is $\delta$–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 Teichmüller space to $\mathcal{C}\left(S\right)$ sending a Riemann surface to the curve(s) of shortest extremal length.