Connected components of the compactification of representation spaces of surface groups

Abstract

The Thurston compactification of Teichmuller spaces has been generalized to many different representation spaces by J. Morgan, P. Shalen, M. Bestvina, F. Paulin, A. 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 is very degenerated: the nice behaviour of the Thurston compactification of the Teichmuller 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 fat R-trees, i.e., R-trees equipped with a planar structure.

75 pages, 11 figures

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Cited by in corpus (3)

• Compact anti-de Sitter 3-manifolds and folded hyperbolic structures on surfaces
• Dynamical rigidity of non discrete representations in PSL(2,R)
• Valuations on the character variety: Newton polytopes and Residual Poisson Bracket