# 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}
}
• 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.
126 Citations

## Figures from this paper

Uniform hyperbolicity of the curve graphs
We show that there is a universal constant, k, such that the curve graph associated to any compact orientable surface is k-hyperbolic. Independent proofs of this have been given by Aougab, by Hensel,
Separating Curve Complex of the Genus Two Surface is Delta Hyperbolic
A proof that the separating curve complex of the closed genus two surface has a quasi-distance formula and is delta hyperbolic using tools of Masur and Schleimer. This answers in the affirmative a
Asymptotic dimension and the disk graph II
We show that the asymptotic dimension of a hyperbolic relatively hyperbolic graph is finite, provided that this holds true uniformly for the peripheral subgraphs and for the electrification. We use
An upper bound on distance degenerate handle additions
We prove that for any distance at least 3 Heegaard splitting and a boundary component $F$, there is a diameter finite ball in the curve complex $\mathcal {C}(F)$ so that it contains all distance
Subsurface distances for hyperbolic 3-manifolds fibering over the circle
• Mathematics
• 2022
For a hyperbolic fibered 3–manifold M , we prove results that uniformly relate the structure of surface projections as one varies the fibrations of M . This extends our previous work from the
A short proof of the bounded geodesic image theorem
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
Asymptotic dimension and the disk graph I
For an aspherical oriented 3‐manifold M and a subsurface X of the boundary of M with empty or incompressible boundary, we use surgery to identify a graph whose vertices are disks with boundary in X
Geometry of graphs of discs in a handlebody
For a handlebody H of genus g>1 we use surgery to identify a graph whose vertices are discs and which is quasi-isometrically embedded in the curve graph of the boundary surface.
Uniform quasiconvexity of the disc graphs in the curve graphs
We give a proof that there exists a universal constant K such that the disc graph associated to a surface S forming a boundary component of a compact, orientable 3-manifold M is K-quasiconvex in the
A note on acylindrical hyperbolicity of Mapping Class Groups
• Mathematics
• 2015
The aim of this note is to give the simplest possible proof that Mapping Class Groups of closed hyperbolic surfaces are acylindrically hyperbolic, and more specifically that their curve graphs are

## 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
• Mathematics
• 2003
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
• Mathematics
• 2007
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