Volume 15, issue 3 (2011)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 20
Issue 4, 1807–2438
Issue 3, 1257–1806
Issue 2, 629–1255
Issue 1, 1–627

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Subscriptions
Author Index
To Appear
Contacts
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Connected components of the compactification of representation spaces of surface groups

Maxime Wolff

Geometry & Topology 15 (2011) 1225–1295
Abstract

The Thurston compactification of Teichmüller spaces has been generalised to many different representation spaces by Morgan, Shalen, Bestvina, Paulin, Parreau and others. In the simplest case of representations of fundamental groups of closed hyperbolic surfaces in PSL(2, ), we prove that this compactification behaves very badly: the nice behaviour of the Thurston compactification of the Teichmüller space contrasts with wild phenomena happening on the boundary of the other connected components of these representation spaces. We prove that it is more natural to consider a refinement of this compactification, which remembers the orientation of the hyperbolic plane. The ideal points of this compactification are oriented –trees, ie, –trees equipped with a planar structure.

Keywords
$\mathbb{R}$–tree, Euler class, surface group, Teichmüller space, Thurston's compactification
Mathematical Subject Classification 2000
Primary: 53C23
Secondary: 20H10, 32G15
References
Publication
Received: 8 August 2008
Revised: 20 April 2011
Accepted: 24 May 2011
Published: 29 July 2011
Proposed: Walter Neumann
Seconded: Ronald J Stern, Danny Calegari
Authors
Maxime Wolff
Institut de Mathématiques de Jussieu
Université Pierre et Marie Curie - Paris 6
Case 247, 4 place Jussieu
Fr-75005 Paris
France
http://www.math.jussieu.fr/~wolff