Connected components of the compactification of representation spaces of surface groups

Wolff, Maxime

doi:10.2140/gt.2011.15.1225

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Connected components of the compactification of representation spaces of surface groups

Maxime Wolff

Geometry & Topology 15 (2011) 1225–1295

DOI: 10.2140/gt.2011.15.1225

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 $PSL (2, ℝ)$ , 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 $ℝ$ –trees, ie, $ℝ$ –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

Volume 15, issue 3 (2011)