Geometry & Topology

Uniform hyperbolicity of the graphs of curves

Tarik Aougab

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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

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

Received: 16 January 2013
Revised: 27 May 2013
Accepted: 3 July 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C12: Distance in graphs 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 57M07: Topological methods in group theory 57M15: Relations with graph theory [See also 05Cxx] 57M20: Two-dimensional complexes

uniform hyperbolicity curve complex mapping class group


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

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