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McShane Identities for Higher Teichmüller Theory and the Goncharov–Shen Potential
About this Title
Yi Huang and Zhe Sun
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 286, Number 1422
ISBNs: 978-1-4704-6312-0 (print); 978-1-4704-7516-1 (online)
DOI: https://doi.org/10.1090/memo/1422
Published electronically: May 16, 2023
Keywords: McShane’s identity,
Fock–Goncharov $\mathcal {A}$ moduli space,
Goncharov–Shen potential
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminary
- 3. Properties of projective invariants
- 4. Goncharov–Shen potentials
- 5. Identities for $\operatorname {PGL}_3(\mathbb {R})$-representations with unipotent boundary
- 6. Simple geodesic sparsity for convex real projective surfaces
- 7. McShane identities for higher Teichmüller space
- 8. Applications
- A. Fuchsian rigidity
- B. More on the length spectrum of cusped strictly convex real projective surfaces
Abstract
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.- Ilesanmi Adeboye and Daryl Cooper, The area of convex projective surfaces and Fock-Goncharov coordinates, J. Topol. Anal. 12 (2020), no. 2, 533–546. MR 4119114, DOI 10.1142/S1793525319500560
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