Bending Teichmüller spaces and character varieties

Abstract

We consider the mapping $b_L:\mathcal{T}\to \ddot{I}‡$ from the Fricke-Teichmüller space $\mathcal{T}$ into the ${\mathrm{PSL}}_2\mathbb{C}$-character variety $\ddot{I}‡$ of the surface, obtained by bending Fuchsian representations along a fixed measured lamination $L$. We prove that this mapping is an equivariant symplectic real-analytic embedding, and, for almost all measured laminations, proper. We also show that this ``bending map'' $b_L:\mathcal{T}\to \ddot{I}‡$ extends continuously almost-everywhere to the canonical inclusion map from the Thurston boundary of $\mathcal{T}$ into the Morgan-Shalen boundary of $\ddot{I}‡$. Moreover, we ``complexify" this bending map in a geometric manner. Namely, we symplectically embed this real-analytic subvariety $\mathrm{Im}{b}_L$ into the product variety $\ddot{I}‡\times \ddot{I}‡$ by the diagonal mapping twisted by complex conjugation. Then we construct a closed $\mathbb{C}$-symplectic complex-analytic subvariety of $\ddot{I}‡\times \ddot{I}‡$ containing $\mathrm{Im}{b}_L$ as a half-dimensional real-analytic subvariety.

53 pages, 11 figures