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Anosov deformations of Barbot representations

Samuel Bronstein, Colin Davalo

TL;DR

The work constructs a real, four‑dimensional slice of the SL(3,\mathbb{R}) character variety around Barbot representations via SL(3,\mathbb{R}) Higgs bundles in a Slodowy slice, proving that all representations in this slice are Borel Anosov and organizing them as holonomies of flag‑space structures. Central to the method is a multicone criterion for Anosovness, implemented by a parallel distribution of planes along a minimal surface in the SL(3,\mathbb{R})/SO(3) symmetric space, together with a maximum principle that bounds the harmonic data. The construction yields a domain of discontinuity in the full flag space that fibers over the surface by conics, with the conic fibration giving a circle‑bundle quotient, and clarifies how the Anosov limit map sits relative to the conics. Altogether, the paper links Barbot deformations to Anosov dynamics and flag‑geometric structures, extending the landscape of discrete, faithful SL(3,\mathbb{R}) representations beyond the Hitchin component.

Abstract

We construct for each conformal structure on a closed orientable surface of genus at least 2 a proper slice in the character variety of representations of the associated surface group into SL(3,R) that belongs to the Barbot component and show that the corresponding representations are Borel Anosov. We describe a fibered geometric structure in the space of full flags associated with these representations.

Anosov deformations of Barbot representations

TL;DR

The work constructs a real, four‑dimensional slice of the SL(3,\mathbb{R}) character variety around Barbot representations via SL(3,\mathbb{R}) Higgs bundles in a Slodowy slice, proving that all representations in this slice are Borel Anosov and organizing them as holonomies of flag‑space structures. Central to the method is a multicone criterion for Anosovness, implemented by a parallel distribution of planes along a minimal surface in the SL(3,\mathbb{R})/SO(3) symmetric space, together with a maximum principle that bounds the harmonic data. The construction yields a domain of discontinuity in the full flag space that fibers over the surface by conics, with the conic fibration giving a circle‑bundle quotient, and clarifies how the Anosov limit map sits relative to the conics. Altogether, the paper links Barbot deformations to Anosov dynamics and flag‑geometric structures, extending the landscape of discrete, faithful SL(3,\mathbb{R}) representations beyond the Hitchin component.

Abstract

We construct for each conformal structure on a closed orientable surface of genus at least 2 a proper slice in the character variety of representations of the associated surface group into SL(3,R) that belongs to the Barbot component and show that the corresponding representations are Borel Anosov. We describe a fibered geometric structure in the space of full flags associated with these representations.

Paper Structure

This paper contains 15 sections, 26 theorems, 75 equations, 1 figure.

Table of Contents

  1. Introduction.
  2. A slice of deformations of Barbot representations.
  3. The non-Abelian Hodge correspondence for SL(3,R).
  4. Deformations of Barbot representations.
  5. The Anosov property.
  6. Multicone criterion
  7. A parallel distribution of planes.
  8. The Higgs bundle, and the parallel distribution of planes.
  9. The maximum principle
  10. Construction of the multicones
  11. Fibration over a reducible hyperbolic plane
  12. Description of the fibration
  13. Geometric structure
  14. Nestedness of the multicones.
  15. Fibration of a domain of discontinuity

Key Result

Theorem 1.1

Representations corresponding to the Higgs bundles $\mathcal{S}_X^\mathbb{R}$ are Anosov.

Figures (1)

  • Figure 1: Illustration of the reducible hyperbolic plane $\mathcal{H}_x$ in the symmetric space

Theorems & Definitions (57)

  • Theorem 1.1: Theorem \ref{['thm:mainText']}
  • Theorem 1.2: Theorem \ref{['thm:Fibration']}
  • Theorem 2.1: Hit87,Cor88,Don03
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 47 more