Skip to main content Accessibility help
×
Home
Hostname: page-component-78dcdb465f-tqmtl Total loading time: 5.719 Render date: 2021-04-14T11:06:29.737Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

On the arc and curve complex of a surface

Published online by Cambridge University Press:  02 December 2009

MUSTAFA KORKMAZ
Affiliation:
Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey. e-mail: korkmaz@metu.edu.tr
ATHANASE PAPADOPOULOS
Affiliation:
Institut de Recherche Mathématique Avancée, Université de Strasbourg and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France, and Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany. e-mail: papadopoulos@math.u-strasbg.fr
Corresponding

Abstract

We study the arc and curve complex AC(S) of an oriented connected surface S of finite type with punctures. We show that if the surface is not a sphere with one, two or three punctures nor a torus with one puncture, then the simplicial automorphism group of AC(S) coincides with the natural image of the extended mapping class group of S in that group. We also show that for any vertex of AC(S), the combinatorial structure of the link of that vertex characterizes the type of a curve or of an arc in S that represents that vertex. We also give a proof of the fact if S is not a sphere with at most three punctures, then the natural embedding of the curve complex of S in AC(S) is a quasi-isometry. The last result, at least under some slightly more restrictive conditions on S, was already known. As a corollary, AC(S) is Gromov-hyperbolic.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2009

Access options

Get access to the full version of this content by using one of the access options below.

References

[1]Hatcher, A.On triangulations of surfaces. Topology Appl. 40 (1991), 189194. A new version is available on the author's webpage.CrossRefGoogle Scholar
[2]Harvey, W. J. Boundary structure of the modular group. Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference, pp. 245251, Ann. of Math. Stud., 97, Princeton Univ. Press, Princeton, NJ, 1981.Google Scholar
[3]Irmak, E. and McCarthy, J. D.Injective simplicial maps of the arc complex, Turkish J. Math. 33 (2009), 116.Google Scholar
[4]Ivanov, N. V. Automorphisms of complexes of curves and of Teichmüller spaces. Internat. Math. Res. Notices 1997, no. 14, 651–666.Google Scholar
[5]Korkmaz, M.Automorphisms of complexes of curves on punctured spheres and on punctured tori. Topology Appl. 95, no. 2 (1999), 85111.CrossRefGoogle Scholar
[6]Luo, F.Automorphisms of the complex of curves. Topology 39, no. 2 (2000), 283298.CrossRefGoogle Scholar
[7]Masur, H. A. and Minsky, Y. N.Geometry of the complex of curves. I. Hyperbolicity. Invent. Math. 138 (1) (1999), 103149.CrossRefGoogle Scholar
[8]Schleimer, S. Notes on the complex of curves. Notes from a mini-course given at Caltech in January 2005, available on the web.Google Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 41 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 14th April 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

On the arc and curve complex of a surface
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

On the arc and curve complex of a surface
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

On the arc and curve complex of a surface
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *