The geometry of the disk complex

@article{Masur2010TheGO,
  title={The geometry of the disk complex},
  author={Howard A. Masur and Saul Schleimer},
  journal={Journal of the American Mathematical Society},
  year={2010},
  volume={26},
  pages={1-62}
}
  • H. Masur, S. Schleimer
  • Published 15 October 2010
  • Mathematics
  • Journal of the American Mathematical Society
We give a distance estimate for the disk complex. We use the distance estimate to prove that the disk complex is Gromov hyperbolic. As another application of our techniques, we find an algorithm which computes the Hempel distance of a Heegaard splitting, up to an error depending only on the genus. 
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References

SHOWING 1-10 OF 69 REFERENCES
Tightness and computing distances in the curve complex
We address the lack of local-finiteness in Harvey’s curve complex by computational means, computing bounds on certain intersection numbers among curves lying on a natural family of geodesics. We give
A Combinatorial Model for the Teichmüller Metric
Abstract.We study how the length and the twisting parameter of a curve change along a Teichmüller geodesic. We then use our results to provide a formula for the Teichmüller distance between two
Intersection numbers and the hyperbolicity of the curve complex
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
Quasiconvexity in the curve complex
Let S be the boundary of a handlebody M. We prove that the set of curves in S that are boundaries of disks in M, considered as a subset of the complex of curves of S, is quasi-convex.
The Geometry of Cycles in the Cayley Diagram of a Group
A study of triangulations of cycles in the Cayley diagrams of finitely generated groups leads to a new geometric characterization of hyperbolic groups.
Geometry of the mapping class groups II: Subsurfaces
Let S be an oriented surface of genus g with m punctures. If 3g-3+m is at least 4 then we construct for every compact subset K of moduli space a closed Teichmueller geodesic not intersecting K.
The classification of Kleinian surface groups I : Models and bounds : preprint
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
Asymptotic Geometry of the Mapping Class Group and Teichmuller Space
In this work, we study the asymptotic geometry of the mapping class group and Teichmuller space. We introduce tools for analyzing the geometry of “projection” maps from these spaces to curve
Covers and curve complex
We provide the first nontrivial examples of quasi-isometric embeddings between curve complexes; these are induced by orbifold covers. This leads to new quasi-isometric embeddings between mapping
Complex Analytic Theory of Teichmuller Spaces
A Portmanteau of Preliminaries The Moduli Spaces For Riemann Surfaces The Complex Structure of Teichmuller Spaces More About the Complex Structure of Teichmuller Spaces The Universal Family and Its
...
1
2
3
4
5
...