Basmajian's identity in higher Teichmüller–Thurston theory

Abstract

We prove an extension of Basmajian's identity to n-Hitchin representations of compact bordered surfaces. For n = 3, we show that this identity has a geometric interpretation for convex real projective structures analogous to Basmajian's original result. As part of our proof, we demonstrate that, with respect to the Lebesgue measure on the Frenet curve associated to a Hitchin representation, the limit set of an incompressible subsurface of a closed surface has measure zero. This generalizes a classical result in hyperbolic geometry. Finally, we recall the Labourie-McShane extension of the McShane-Mirzakhani identity to Hitchin representations and note a close connection to Basmajian's identity in both the hyperbolic and the Hitchin settings.

Homéomorphismes et nombre d'intersection

“…In their paper [127], they extend Thurston's shear coordinates to the setting of Hitchin representations of fundamental groups of surfaces and they prove a McShane-Mirzakhani identity in that setting. Vlamis and Yarmola use the same expression in the paper where they prove a Basmajian identity in higher Teichmüller-Thurston theory [236]. Among the large number of results in higher Teichmüller theory that are inspired by Thurston's work on surfaces, we mention Labourie's work on representations of surface fundamental groups into PSL(n, R), and in particular his discovery of a curve which is the limit set of the quasi-Fuchsian representation in this setting [126].…”

Identities for hyperconvex Anosov representations

“…In this section, we generalize Basmajian's identity for hyperbolic manifolds to (1, 1, 2)-hyperconvex Anosov representations ρ : π 1 Σ → PGL(n, R). If ρ is Hithcin, we recover Vlamis-Yarmola's identity ( [11]). Theorem 3.1.…”

“…Proof. Vlamis-Yarmola's proof ( [11], section 3) works verbatim here. The property of the limit set used in their proof is that it is a C 1+α -submanifold of P(R n ).…”

“…where ρ is a notion of length with respect to ρ, C ρ is a cross ratio defined for four points on the boundary at infinity ∂π 1 Σ of the hyperbolic group π 1 Σ and α ± 1 are the (attracting and repelling) fixed points of [∂Σ] on ∂π 1 Σ. Moreover, if ρ is Hitchin (which is (1, 1, 2)-hyperconvex), equation (1.2) recovers Vlamis-Yarmola's identity [11].…”

A Glimpse into Thurston’s Work

“…In their paper [127], they extend Thurston's shear coordinates to the setting of Hitchin representations of fundamental groups of surfaces and they prove a McShane-Mirzakhani identity in that setting. Vlamis and Yarmola use the same expression in the paper where they prove a Basmajian identity in higher Teichmüller-Thurston theory [238]. Among the large number of results in higher Teichmüller theory that are inspired by Thurston's work on surfaces, we mention Labourie's work on representations of surface fundamental groups into PSL(n, R), and in particular his discovery of a curve which is the limit set of the quasi-Fuchsian representation in this setting [126].…”