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Ghost polygons, Poisson bracket and convexity

Martin Bridgeman, François Labourie

TL;DR

The paper develops a nonabelian calculus for natural functions on deformation spaces of Anosov representations by introducing ghost polygons and a ghost algebra to compute derivatives of correlation functions. It proves a Poisson-bracket stability: the Poisson bracket of length and correlation functions can be expressed combinatorially via ghost intersections, enabling recursive bracket computations. A key advance is the uniform framework of uniformly hyperbolic bundles, including a fundamental projector with a cohomological equation governing its variation, which underpins ghost integration and intersection. In applications, positivity yields convexity of averaged length functions, and the construction of commuting subalgebras arises from triangle- and lamination-associated functions; the theory extends to closed surfaces through cyclic currents and averaging procedures. Overall, the work generalizes Goldman-type Poisson brackets to higher-rank, nonperiodic settings and provides a robust toolkit for analyzing Hamiltonian dynamics of length and correlation data in Teichmüller-type spaces.

Abstract

The moduli space of Anosov representations of a surface group in a semisimple group, which is an open set in the character variety, admits many more natural functions than the regular functions. We will study in particular length functions and, correlation functions. Our main result is a formula that computes the Poisson bracket of those functions using some combinatorial devices called {\em ghost polygons} and {\em ghost bracket} encoded in a formal algebra called {\em ghost algebra} related in some cases to the swapping algebra introduced by the second author. As a consequence of our main theorem, we show that the set of those functions -- length and correlation -- is stable under the Poisson bracket. We give two applications: firstly in the presence of positivity we prove the convexity of length functions, generalising a result of Kerckhoff in Teichmüller space, secondly we exhibit subalgebras of commuting functions. An important tool is the study of {\em uniformly hyperbolic bundles} which is a generalisation of Anosov representations beyond periodicity.

Ghost polygons, Poisson bracket and convexity

TL;DR

The paper develops a nonabelian calculus for natural functions on deformation spaces of Anosov representations by introducing ghost polygons and a ghost algebra to compute derivatives of correlation functions. It proves a Poisson-bracket stability: the Poisson bracket of length and correlation functions can be expressed combinatorially via ghost intersections, enabling recursive bracket computations. A key advance is the uniform framework of uniformly hyperbolic bundles, including a fundamental projector with a cohomological equation governing its variation, which underpins ghost integration and intersection. In applications, positivity yields convexity of averaged length functions, and the construction of commuting subalgebras arises from triangle- and lamination-associated functions; the theory extends to closed surfaces through cyclic currents and averaging procedures. Overall, the work generalizes Goldman-type Poisson brackets to higher-rank, nonperiodic settings and provides a robust toolkit for analyzing Hamiltonian dynamics of length and correlation data in Teichmüller-type spaces.

Abstract

The moduli space of Anosov representations of a surface group in a semisimple group, which is an open set in the character variety, admits many more natural functions than the regular functions. We will study in particular length functions and, correlation functions. Our main result is a formula that computes the Poisson bracket of those functions using some combinatorial devices called {\em ghost polygons} and {\em ghost bracket} encoded in a formal algebra called {\em ghost algebra} related in some cases to the swapping algebra introduced by the second author. As a consequence of our main theorem, we show that the set of those functions -- length and correlation -- is stable under the Poisson bracket. We give two applications: firstly in the presence of positivity we prove the convexity of length functions, generalising a result of Kerckhoff in Teichmüller space, secondly we exhibit subalgebras of commuting functions. An important tool is the study of {\em uniformly hyperbolic bundles} which is a generalisation of Anosov representations beyond periodicity.
Paper Structure (102 sections, 89 theorems, 339 equations, 5 figures)

This paper contains 102 sections, 89 theorems, 339 equations, 5 figures.

Table of Contents

  1. Preliminary
  2. The hyperbolic plane, geodesics and forms
  3. Geodesics and intersection
  4. Geodesic forms
  5. The generic set and barycentric construction
  6. Geodesically bounded forms
  7. Polygonal arcs form
  8. The Goldman symplectic form
  9. On the hyperbolic plane
  10. Uniformly hyperbolic bundles and projectors
  11. Uniformly hyperbolic bundles: definitions
  12. Metrics on Grassmannian
  13. Uniformly hyperbolic bundles: definitions
  14. Limit maps
  15. Families of uniformly hyperbolic bundles and their variations
  16. ...and 87 more sections

Key Result

Theorem A

The space of length functions and correlation functions is stable under the Poisson bracket. More precisely there exists a Lie bracket on the polynomial algebra formally generated by tuples of geodesics $(G,H)\mapsto [G,H]$ so that

Figures (5)

  • Figure 1: Two ways to see a cyclically ordered tuple of geodesics
  • Figure 2: Curves $\gamma_i^{\pm}$ and $\eta_i$
  • Figure 3: Appearance of ghosts
  • Figure 4: Polygonal arc ${\eta}_i$ for ghost edge $\zeta_i$
  • Figure 5: Curves $\gamma_0$ and $\gamma_1$

Theorems & Definitions (171)

  • Theorem A: Poisson stability
  • Theorem B: Convexity Theorem
  • Theorem C: Commuting subalgebra
  • Definition 1.1.1
  • Definition 1.1.2
  • Proposition 1.1.3
  • proof
  • Proposition 1.1.4
  • proof
  • Proposition 1.1.5
  • ...and 161 more