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Basmajian's identity in higher Teichmüller-Thurston theory

Nicholas G. Vlamis, Andrew Yarmola

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.

Basmajian's identity in higher Teichmüller-Thurston theory

Abstract

We prove an extension of Basmajian's identity to -Hitchin representations of compact bordered surfaces. For , 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.

Paper Structure

This paper contains 17 sections, 19 theorems, 86 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Hitchin representations
  4. Doubling a Hitchin represenation
  5. The boundary at infinity
  6. The Frenet Curve
  7. Cross Ratios
  8. Lebesgue Measure on the Frenet Curve
  9. Orthogeodesics and Double Cosets
  10. Real Projective Structures $(n = 3)$
  11. Hilbert metric
  12. Basmajian's identity
  13. Basmajian's Identity
  14. Relations to the McShane-Mirzakhani Identity
  15. McShane-Mirzakhani Decomposition
  16. ...and 2 more sections

Key Result

Theorem 1.1

(Basmajian's identity for Hitchin Representations) Let $\Sigma$ be an oriented compact connected surface with $m>0$ boundary components whose double has genus at least 2. Let $\mathcal{A}=\{\alpha_1, \ldots, \alpha_m\}$ be a positive peripheral marking. If $\rho$ is a Hitchin representation of $\pi_ where $\ell_\rho(\partial \Sigma) = \sum_{i=1}^m \ell_\rho(\alpha_i).$ Furthermore, if $\rho$ is Fu

Figures (4)

  • Figure 1: Orthogonal projection of $g\cdot L_j$ onto $L_i$ whose image we defined as $\widetilde{U}^g_{i,j}$.
  • Figure 2: A standard diagram for the orthogonal projection of $g\cdot L_j$ onto $L_i$ in the hyperbolic case.
  • Figure 3: An example of $P_x$ with $\delta_x$ non-simple.
  • Figure 4: A fundamental domain $D$ for $P$ and the lift of $\delta_x$. One can verify that $\alpha \gamma \beta = e$ and $\alpha^{-1} \cdot \beta^\pm = \gamma \cdot \beta^\pm$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: labourie1
  • Corollary 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 21 more