Connected components of the compactification of representation spaces of surface groups

Details Connected components of the compactification of representation spaces of surface groups Connected components of the compactification of representation spaces of surface groups

2008-02-04

arxiv.org/abs/0802.0512v1

Mathematics

Geometric Topology

75 pages, 11 figures

Scientific paper

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.

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