Geometric structures and surface group representations into Lie groups
Final Report Abstract
Let S be a closed oriented surface of genus at least two. During the funding period, the following result has been accepted and published. Considered all CP1 -structure on S with fixed holonomy representation π1 (S) → PSL(2, C). A grafting is a certain cut-and-paste operation which transforms a CP1 -structure to a different CP1-structure, preserving its holonomy representation. In the published paper, for a generic holonomy ρ : π1 (S) → PSL(2, C), I have shown that grafting produces all CP1 with ρ. I have worked on a certain degeneration of CP1-structures on S. Consider a path of CP1-structures Ct leaving every compact subset in the deformation of CP1-structures on S such that its holonomy converges in the PSL(2, C)-character variety. In this setting, it is known that the complex structure Xt of Ct also diverges to infinity in the Teichmüller space. Under the assumption that Xt is pinched along a single loop, I gave descriptions of the asymptotic behavior of Ct in terms of its holomorphic quadratic differential, developing map, and pleated surface. During the funded period of time, I have partially written up the result.
