McShane identities for Higher Teichmüller theory and the Goncharov-Shen potential

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.

107 pages, 27 figures. To appear in Mem. Amer. Math. Soc

References in corpus (7)

• Minimal stretch maps between hyperbolic surfaces
• A variation of McShane's identity for 2-bridge links
• Polynomial automorphisms of C^n preserving the Markoff-Hurwitz polynomial
• Super McShane identity
• McShane-type identities for quasifuchsian representations of nonorientable surfaces
• Primitive stable representations in higher rank semisimple Lie groups
• Volume of the moduli space of unmarked bounded positive convex ${\mathbb{RP}}^2$ structures

Cited by in corpus (9)

• Tropical Fock-Goncharov coordinates for ${\mathrm{SL}}_3$-webs on surfaces I: construction
• Super McShane identity
• Optimal Lipschitz Maps on One-holed Tori and the Thurston Metric Theory of Teichmueller Space
• Geodesic and orthogeodesic identities on hyperbolic surfaces
• Primitive stable representations in higher rank semisimple Lie groups
• Tropical Fock-Goncharov coordinates for ${\mathrm{SL}}_3$-webs on surfaces II: naturality
• Volume of the moduli space of unmarked bounded positive convex ${\mathbb{RP}}^2$ structures
• Ideal triangles, hyperbolic surfaces and the Thurston metric on Teichm{ü}ller space
• Minimal stretch maps between Euclidean triangle?