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Boundary Currents of Hitchin Components

Charles Reid

TL;DR

This work extends the Thurston-style boundary of Teichmüller space to Hitchin components $Hit^n(S)$ by pairing spectral data with geodesic currents. It introduces the spectral radius compactification and a geometric boundary via tropical rank $n$ currents, together with a universal asymmetric dual space $X_\mu$ that encodes the asymptotics of length spectra through periods and holonomies. For discrete tropical rank-$n$ currents, $X_\mu$ is a polyhedral complex of dimension at most $n-1$, while in the rank-2 case it recovers the familiar $\mathbb{R}$-tree boundary; in the cubic-differential case ($n=3$) endpoints correspond to a triangular Finsler metric on $\tilde{S}$, with $X_\mu$ isometric to $\tilde{S}$ under the symmetrized metric. The paper develops a rich framework connecting holonomy, curvature, and currents, including a relative metric construction over $X_\mu$, horofunction-based Liouville currents, and a detailed analysis of cubic differential rays, thereby advancing a geometric realization of infinity for higher Teichmüller spaces. This provides a robust toolset for understanding asymptotic geometry and length-spectrum data in Hitchin components and suggests broad avenues for extending these ideas to other Lie groups and compactifications.

Abstract

The space of Hitchin representations of the fundamental group of a closed surface $S$ into $\text{SL}_n\mathbb{R}$ embeds naturally in the space of projective oriented geodesic currents on $S$. We find that currents in the boundary have combinatorial restrictions on self-intersection which depend on $n$. We define a notion of dual space to an oriented geodesic current, and show that the dual space of a discrete boundary current of the $\text{SL}_n\mathbb{R}$ Hitchin component is a polyhedral complex of dimension at most $n-1$. For endpoints of cubic differential rays in the $\text{SL}_3\mathbb{R}$ Hitchin component, the dual space is the universal cover of $S$, equipped with an asymmetric Finsler metric which records growth rates of trace functions.

Boundary Currents of Hitchin Components

TL;DR

This work extends the Thurston-style boundary of Teichmüller space to Hitchin components by pairing spectral data with geodesic currents. It introduces the spectral radius compactification and a geometric boundary via tropical rank currents, together with a universal asymmetric dual space that encodes the asymptotics of length spectra through periods and holonomies. For discrete tropical rank- currents, is a polyhedral complex of dimension at most , while in the rank-2 case it recovers the familiar -tree boundary; in the cubic-differential case () endpoints correspond to a triangular Finsler metric on , with isometric to under the symmetrized metric. The paper develops a rich framework connecting holonomy, curvature, and currents, including a relative metric construction over , horofunction-based Liouville currents, and a detailed analysis of cubic differential rays, thereby advancing a geometric realization of infinity for higher Teichmüller spaces. This provides a robust toolset for understanding asymptotic geometry and length-spectrum data in Hitchin components and suggests broad avenues for extending these ideas to other Lie groups and compactifications.

Abstract

The space of Hitchin representations of the fundamental group of a closed surface into embeds naturally in the space of projective oriented geodesic currents on . We find that currents in the boundary have combinatorial restrictions on self-intersection which depend on . We define a notion of dual space to an oriented geodesic current, and show that the dual space of a discrete boundary current of the Hitchin component is a polyhedral complex of dimension at most . For endpoints of cubic differential rays in the Hitchin component, the dual space is the universal cover of , equipped with an asymmetric Finsler metric which records growth rates of trace functions.

Paper Structure

This paper contains 36 sections, 61 theorems, 150 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Spectral radius compactification
  3. Geodesic currents at infinity
  4. Geometry at infinity
  5. Heuristic description of $X_\mu$
  6. Future Directions
  7. Acknowledgments
  8. Equivariant bundles and geodesic currents
  9. Taxi connections
  10. Geodesic currents and holonomy functions
  11. Holonomy functions and equivariant bundles
  12. Geodesic currents and length spectra
  13. Defining periods
  14. Reduced length spectra and currents
  15. Rank $n$ and tropical rank $n$ holonomy functions
  16. ...and 21 more sections

Key Result

Theorem 1

If $[\mu]\in \mathbb{P}(\mathcal{C}(S))$ is in the boundary of $\mathop{\mathrm{Hit}}\nolimits^n(S)$, then there can not be $(x_1,y_1),...,(x_n,y_n)\in \mathop{\mathrm{supp}}\nolimits(\mu)$, with $x_1 < \dots < x_n < y_1 <\dots < y_n$.

Figures (19)

  • Figure 1.1: A forbidden configuration for a current in the boundary of $\mathop{\mathrm{Hit}}\nolimits^3(S)$
  • Figure 1.2: Examples of universal symmetric dual spaces to some unoriented geodesic currents on the disk. The first two examples are no different from the dual spaces defined in BIPP21, but there is a difference in the third example.
  • Figure 1.3: Examples of $X_\mu$ for some oriented geodesic currents on the disk, with holonomy functions. Generally, when $\mu$ is symmetric, $X_\mu$ is bigger than $X_\mu^{sym}$, but when $\mu$ is a measured lamination, $X_\mu = X_\mu^{sym}$.
  • Figure 2.1: Area between a taxi-loop winding around $\mathcal{G}$ and its $\gamma$ translate
  • Figure 3.1: Comparing the two paths connecting $R_N$ to $R_E$
  • ...and 14 more figures

Theorems & Definitions (141)

  • Definition 1.1
  • Theorem 1
  • Theorem 2: Lemma \ref{['Xmu is dimension n-1']}
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • ...and 131 more