Geometry & Topology

Connected components of the compactification of representation spaces of surface groups

Maxime Wolff

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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.

Article information

Geom. Topol., Volume 15, Number 3 (2011), 1225-1295.

Received: 8 August 2008
Revised: 20 April 2011
Accepted: 24 May 2011
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
Secondary: 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx] 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]

$\mathbb{R}$–tree Euler class surface group Teichmüller space Thurston's compactification


Wolff, Maxime. Connected components of the compactification of representation spaces of surface groups. Geom. Topol. 15 (2011), no. 3, 1225--1295. doi:10.2140/gt.2011.15.1225.

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