# 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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## References

SHOWING 1-10 OF 25 REFERENCES

### Uniform hyperbolicity of the graphs of curves

- Mathematics
- 2012

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 i–hyperbolic for some i D i.g;p/ .…

### Minimally intersecting filling pairs on surfaces

- Mathematics
- 2015

Let Sg denote the closed orientable surface of genus g . We construct exponentially many mapping class group orbits of pairs of simple closed curves which fill Sg and intersect minimally, by showing…

### Geometry of the mapping class groups III: Quasi-isometric rigidity

- Mathematics
- 2005

Let S be an oriented surface of finite type of genus g with m punctures and where 3g-3+m>1. We show that the mapping class group M(S) of S is quasi-isometrically rigid. We also give a different proof…

### Intersection numbers and the hyperbolicity of the curve complex

- Mathematics
- 2006

Abstract We give another proof of the result of Masur and Minsky that the complex of curves associated to a compact orientable surface is hyperbolic. Our proof is more combinatorial in nature and can…

### THE CLASSIFICATION OF KLEINIAN SURFACE GROUPS, II: THE ENDING LAMINATION CONJECTURE

- Mathematics
- 2004

Thurston’s Ending Lamination Conjecture states that a hyperbolic 3manifold N with nitely generated fundamental group is uniquely determined by its topological type and its end invariants. In this…

### The classification of Kleinian surface groups I : Models and bounds : preprint

- Mathematics
- 2002

We give the first part of a proof of Thurston’s Ending Lamination conjecture. In this part we show how to construct from the end invariants of a Kleinian surface group a “Lipschitz model” for the…

### A Representation of Orientable Combinatorial 3-Manifolds

- Mathematics
- 1962

The following question has been posed by Bing [1]: "Which compact, connected 3-manifolds can be obtained from S3 as follows: Remove a finite collection of mutually exclusive (but perhaps knotted and…

### Geometry and rigidity of mapping class groups

- Mathematics
- 2008

We study the large scale geometry of mapping class groups MCG(S), using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of MCG(S) (outside a few sporadic cases) is a…

### A short proof of the bounded geodesic image theorem

- Mathematics
- 2013

We give a combinatorial proof, using the hyperbolicity of the curve graphs, of the bounded geodesic image theorem of Masur and Minsky. Recently it has been shown that curve graphs are uniformly…