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

Abstract

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) McShanetype 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.

From hyperbolic Dehn filling to surgeries in representation varieties

“…Generalizations of the McShane identities for higher Teichmüller spaces were obtained by Y. Huang and Z. Sun in [47]; these are expressed in terms of simple root lengths, triple ratios and edge functions. I.…”

Hitchin representations of Fuchsian groups

“…(4) François Labourie and Greg McShane [66] proved an analogue of McShane's identity for Hitchin representations, while Nick Vlamis and Andrew Yarmola [120] proved an analogue of the Basmajian identity for Hitchin representations. Yi Huang and Zhe Sun [55] proved versions of McShane's identity for holonomy maps of finite area convex projective surfaces (and for positive representations of Fuchsian lattices). ( 5) Richard Schwartz and Richard Sharp [108] proved an explicit correlation result for lengths on hyperbolic surfaces.…”

The infinitesimal and global Thurston geometry of Teichm{ü}ller space

“…Now cover each P i with two disjoint (except along their boundaries) ideal triangles 1 i , 2 i . We invoke the fact that the collection of embedded ideal triangles on S is compact [10,Prop. 3.8], and by changing indices we produce { 1 i } i∈N which converges to an embedded ideal triangle 1 .…”