Reference : McShane identities for Higher Teichmüller theory and the Goncharov-Shen potential
Scientific journals : Article
Physical, chemical, mathematical & earth Sciences : Mathematics
McShane identities for Higher Teichmüller theory and the Goncharov-Shen potential
Sun, Zhe mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Huang, Yi mailto [Yau Mathematical Sciences Center]
Memoirs of the American Mathematical Society
American Mathematical Society
United States
[en] Mcshane's identity ; Fock–Goncharov A moduli space ; Goncharov-Shen potential
[en] 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.
FnR ; FNR13242285 > Zhe Sun > > COmbinatorial and ALgebraic Aspects of Surface group representations. > 01/09/2018 > 31/08/2020 > 2017

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