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

Yi Huang Yau Mathematical Sciences Center Zhe Sun University of Luxembourg

Geometric Analysis and Geometric Topology Representation Theory mathscidoc:1905.15001

89, 2019.1
In [GS15], Goncharov and Shen introduce a family of mapping class group invariant regular functions on their A-moduli space to explicitly formulate a particular homological mirror symmetry conjecture. Using these regular functions, we obtain McShane identities general rank positive surface group representations with loxodromic boundary monodromy and (non-strict) McShane-type inequalities for general rank positive representations with unipotent boundary monodromy. Our identities are expressed in terms of projective invariants, and we study these invariants: we establish boundedness and Fuchsian rigidity results for triple ratios. Moreover, we obtain McShane identities for finite-area cusped convex real projective surfaces by generalizing the Birman--Series geodesic scarcity theorem. We apply our identities to derive the simple spectral discreteness of unipotent bordered positive representations, collar lemmas, and generalizations of the Thurston metric.
Mcshane's identity, Fock–Goncharov A moduli space, Goncharov-Shen potential
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  • arXiv preprint. 89 pages, 25 figures. Version before submission. Comments welcome
  title={McShane identities for Higher Teichmüller theory and the Goncharov-Shen potential},
  author={Yi Huang, and Zhe Sun},
Yi Huang, and Zhe Sun. McShane identities for Higher Teichmüller theory and the Goncharov-Shen potential. 2019. pp.89.
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