2022/03/29 by Shinpei Baba, Baba, Shinpei
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2203.15394
openalex publication_date 2022/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We consider the mapping bL\colonT → χ from the Fricke-Teichmüller space T into the PSL2ℂ-character variety χ 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'' bL\colon T → χ extends continuously almost-everywhere to the canonical inclusion map from the Thurston boundary of T into the Morgan-Shalen boundary of χ. Moreover, we ``complexify" this bending map in a geometric manner. Namely, we symplectically embed this real-analytic subvariety \rm Im bL into the product variety χ× χ by the diagonal mapping twisted by complex conjugation. Then we construct a closed ℂ-symplectic complex-analytic subvariety of χ× χ containing Im bL as a half-dimensional real-analytic subvariety.