#### Volume 17, issue 5 (2013)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
Uniform hyperbolicity of the graphs of curves

### Tarik Aougab

Geometry & Topology 17 (2013) 2855–2875
##### Abstract

Let $\mathsc{C}\left({S}_{g,p}\right)$ denote the curve complex of the closed orientable surface of genus $g$ with $p$ punctures. Masur and Minksy and subsequently Bowditch showed that $\mathsc{C}\left({S}_{g,p}\right)$ 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 ${\mathsc{C}}_{1}\left({S}_{g,p}\right)$ 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 $\mathsc{C}\left(S\right)$ sending a Riemann surface to the curve(s) of shortest extremal length.

##### Keywords
uniform hyperbolicity, curve complex, mapping class group
##### Mathematical Subject Classification 2010
Primary: 05C12, 20F65, 57M07, 57M15, 57M20