#### Volume 15, issue 3 (2011)

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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 $PSL\left(2,ℝ\right)$, 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