## Geometry & Topology

### Uniform hyperbolicity of the graphs of curves

Tarik Aougab

#### 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 $δ$–hyperbolic for some $δ=δ(g,p)$. In this paper, we show that there exists some $δ>0$ independent of $g,p$ such that the curve graph $C1(Sg,p)$ is $δ$–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 $C(S)$ sending a Riemann surface to the curve(s) of shortest extremal length.

#### Article information

Source
Geom. Topol., Volume 17, Number 5 (2013), 2855-2875.

Dates
Revised: 27 May 2013
Accepted: 3 July 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732689

Digital Object Identifier
doi:10.2140/gt.2013.17.2855

Mathematical Reviews number (MathSciNet)
MR3190300

Zentralblatt MATH identifier
1273.05050

#### Citation

Aougab, Tarik. Uniform hyperbolicity of the graphs of curves. Geom. Topol. 17 (2013), no. 5, 2855--2875. doi:10.2140/gt.2013.17.2855. https://projecteuclid.org/euclid.gt/1513732689

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