Connected components of the compactification of representation spaces of surface groups

TL;DR

This paper investigates the structure of the compactification of representation spaces of surface groups, revealing degenerate behaviors and proposing a refined compactification that captures additional geometric information.

Contribution

It introduces a refined compactification for surface group representations that accounts for orientation, leading to a better understanding of boundary phenomena and ideal points as fat R-trees.

Findings

The standard Thurston compactification is degenerate for certain representation spaces.

A refined compactification considering orientation provides a more natural boundary description.

Ideal points are characterized as fat R-trees with planar structures.

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.