# Small intersection numbers in the curve graph

@article{Aougab2014SmallIN, title={Small intersection numbers in the curve graph}, author={Tarik Aougab and Samuel J. Taylor}, journal={Bulletin of the London Mathematical Society}, year={2014}, volume={46} }

Let Sg,p denote the genus g orientable surface with p⩾0 punctures, and let ω(g,p)=3g+p−3>1 . We prove the existence of infinitely long geodesic rays (v0,v1,v2,…) in the curve graph satisfying the following optimal intersection property: for any natural numbers i and k , the endpoints vi,vi+k of any length k subsegment intersect at most fi,k(ω) times, where fi,k(x) is O(xk−2) . This answers a question of Dan Margalit.

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