Bending Teichmüller spaces and character varieties

Abstract: We consider the mapping $b_L\colon\mathcal{T} \to \chi$ of the Fricke-Teichm\"uller space $\mathcal{T}$ into the $\mathrm{PSL}_2\mathbb{C}$-character variety $\chi$ of the surface, obtained by holonomy representations of bent hyperbolic surfaces 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. In addition, we show that this "being map'' $b_L\colon \mathcal{T} \to \chi$ continuously extends to a mapping from Thurston's boundary of $\mathcal{T}$ to the Morgan-Shalen boundary of $\chi$ as the identity map almost everywhere. Moreover, we complexify the real analytic subvariety ${\rm Im}\, b_L$ after symplecitcaly embedding it in the product variety $\chi \times \chi$ by the diagonal mapping twisted by complex conjugation. More precisely, we geometrically construct a closed $\mathbb{C}$-symplectic complex analytic subvariety of $\chi \times \chi$ containing ${\rm Im}\, b_L$ as a half-dimensional real analytic subvariety.