Bending Teichm\"uller spaces and character varieties

TL;DR

This paper studies the properties of the bending map from Teichmüller space to character varieties, proving it is a symplectic embedding, extending to boundaries, and complexifying it within a geometric framework.

Contribution

It establishes the symplectic real-analytic nature of the bending map, its boundary extension, and introduces a geometric complexification within character varieties.

Findings

The bending map is an equivariant symplectic real-analytic embedding.

The bending map extends continuously to the Thurston boundary.

A geometric complexification of the bending map is constructed within character varieties.

Abstract

We consider the mapping $b_L\colon\mathcal{T} \to \chi$ from the Fricke-Teichm\"uller space $\mathcal{T}$ into the $\mathrm{PSL}_2\mathbb{C}$-character variety $\chi$ 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'' $b_L\colon \mathcal{T} \to \chi$ extends continuously almost-everywhere to the canonical inclusion map from the Thurston boundary of $\mathcal{T}$ into the Morgan-Shalen boundary of $\chi$. Moreover, we ``complexify" this bending map in a geometric manner. Namely, we symplectically embed this real-analytic subvariety ${\rm Im} b_L$ into the product variety $\chi \times \chi$ by the diagonal mapping twisted by complex conjugation. Then we construct a closed $\mathbb{C}$-symplectic complex-analytic subvariety of $\chi \times \chi$ containing $\mathrm{Im} b_L$ as a half-dimensional real-analytic subvariety.