Basmajian's identity in higher Teichm\"uller-Thurston theory

TL;DR

This paper extends Basmajian's identity to higher Teichmüller-Thurston theory, specifically for n-Hitchin representations, and explores its geometric interpretation and measure-theoretic properties, generalizing classical hyperbolic geometry results.

Contribution

It introduces a generalized Basmajian's identity for n-Hitchin representations and connects it to convex real projective structures and measure-zero limit sets.

Findings

Extended Basmajian's identity to n-Hitchin representations

Identified geometric interpretation for convex real projective structures when n=3

Proved limit set of an incompressible subsurface has measure zero

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.