The Teichmuller space of a hyperbolic surface can be identified with a connected component of the character variety of surface group representations into the Lie group PSL(2,R). Thurston provided a compactification of this space and studied the action of the mapping class group on the resulting closed ball. The boundary points can be described as projectivized measured laminations or actions on R-trees.

Higher Teichmuller theory is (in part) concerned with preferred connected components of character varieties of surface group representations into higher rank real semisimple Lie groups. In this talk we will focus on the Lie group PSL(2,R) x PSL(2,R). We show that Ouyang's compactification of the corresponding higher Teichmuller component via cores of trees is a closed ball and study the action of the mapping class group on the resulting space. We describe the boundary points as vector valued mixed structures which, we conjecture, should arise in compactifications of the higher Teichmuller components of other rank 2 Lie groups.

This is joint work with Charles Ouyang and Andrea Tamburelli. Speaker(s): Giuseppe Martone (University of Michigan)