eBook

McShane Identities for Higher Teichmüller Theory and the Goncharov–Shen Potential

Sun, Zhe, Huang, Yi

2023

Overview

We derive generalizations of McShane’s identity for higher ranked surface group representations by studying a family of mapping class group invariant functions introduced by Goncharov and Shen, which generalize the notion of horocycle lengths. In particular, we obtain McShane-type identities for finite-area cusped convex real projective surfaces by generalizing the Birman–Series geodesic scarcity theorem. More generally, we establish McShane-type identities for positive surface group representations with loxodromic boundary monodromy, as well as McShane-type inequalities for general rank positive representations with unipotent boundary monodromy. Our identities are systematically expressed in terms of projective invariants, and we study these invariants: we establish boundedness and Fuchsian rigidity results for triple and cross ratios. We apply our identities to derive the simple spectral discreteness of unipotent-bordered positive representations, collar lemmas, and generalizations of the Thurston metric.

• Publisher: American Mathematical Society
• Subject: Combinatorial identities / Convex surfaces / Cusp forms (Mathematics) / Geodesics (Mathematics) / Low-dimensional topology / Manifolds and cell complexes -- Low-dimensional topology -- Geometric structures on low-dimensional manifolds msc / Potential theory (Mathematics) / Several complex variables and analytic spaces -- Deformations of analytic structures -- Moduli of Riemann surfaces, Teichmüller theory msc / Teichmu¨ller spaces / Teichmüller spaces
• ISBN: 1470463121, 9781470463120