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On Geometric Structures in the Einstein Universe for $\mathrm{SO}_0(p,p+1)$-Hitchin Representations

Colin Davalo, Parker Evans

TL;DR

This work extends the understanding of fiber topology for domains of discontinuity associated with Hitchin representations to the family (SO_0(p,p+1), Ein^{p-1,p}) for p ≥ 3. By combining Higgs-bundle/Data with a pencil-based base analysis and a deformation strategy, the authors show that the quotient M_p → S has fibers that are Ein^{p-1,p-2} when p is odd and T^1 RP^{p-1} when p is even (with exceptional cases p ∈ {4,8} where the tangent bundle is trivial and fibers coincide with Ein^{p-1,p-2}). They implement a robust framework based on Ein-regular pencils, the base-bundle E → RP^{p-1}, and a Nearest-Point Projection technique (via Dav25) to compute the fiber topology, and provide corollaries for G_2'-Hitchin representations. The results deliver new explicit fiber-topology descriptions, enriching the geometric understanding of (G,X)-structures arising from Hitchin components and their G_2-inclusions in higher rank settings.

Abstract

Let $S$ be a closed surface of genus $g \geq 2$. We determine the topology of the fibers of the domain of discontinuity in $\mathrm{Ein}^{p-1,p}$ defined by Guichard-Wienhard and Kapovich-Leeb-Porti for $\mathrm{SO}_0(p,p+1)$-Hitchin representations for $p \geq 3$.

On Geometric Structures in the Einstein Universe for $\mathrm{SO}_0(p,p+1)$-Hitchin Representations

TL;DR

This work extends the understanding of fiber topology for domains of discontinuity associated with Hitchin representations to the family (SO_0(p,p+1), Ein^{p-1,p}) for p ≥ 3. By combining Higgs-bundle/Data with a pencil-based base analysis and a deformation strategy, the authors show that the quotient M_p → S has fibers that are Ein^{p-1,p-2} when p is odd and T^1 RP^{p-1} when p is even (with exceptional cases p ∈ {4,8} where the tangent bundle is trivial and fibers coincide with Ein^{p-1,p-2}). They implement a robust framework based on Ein-regular pencils, the base-bundle E → RP^{p-1}, and a Nearest-Point Projection technique (via Dav25) to compute the fiber topology, and provide corollaries for G_2'-Hitchin representations. The results deliver new explicit fiber-topology descriptions, enriching the geometric understanding of (G,X)-structures arising from Hitchin components and their G_2-inclusions in higher rank settings.

Abstract

Let be a closed surface of genus . We determine the topology of the fibers of the domain of discontinuity in defined by Guichard-Wienhard and Kapovich-Leeb-Porti for -Hitchin representations for .
Paper Structure (11 sections, 17 theorems, 41 equations, 2 figures)

This paper contains 11 sections, 17 theorems, 41 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Model Space Ein(p-1,p) in the Visual Boundary
  4. Hitchin Representations
  5. Higgs Bundles for Fuchsian-Hitchin Representations
  6. Domains of Discontinuity via Tits Metric Thickening
  7. Domains of Discontinuity via Bases of Pencils
  8. (SO(p,p+1),Ein(p-1,p))-Geometric Structures
  9. Geometry of the Ein-Base
  10. The Odd Case: Fibers
  11. The Even Case: Fibers

Key Result

Theorem 1.1

Let $p \geq 3$ be an integer and $\rho:\pi_1S \rightarrow \mathrm{SO}_0(p,p+1)$ a Hitchin representation. The quotient $\rho(\pi_1S)\backslash \Omega_\rho$ has the following smooth fibers:

Figures (2)

  • Figure 1: The pencil $\mathfrak{E_t}$ visualized diagrammatically on $\mathcal{E}$ in the case $p =3$. The forwards arrows come from $\Phi_t$ and the backwards arrows are contributed by $\Phi_t^{*h}$.
  • Figure 2: The pencil $\mathfrak{E_t}$ visualized diagrammatically on $\mathcal{E}$ in the case $p =4$. The forwards arrows come from $\Phi_t$ and the backwards arrows are contributed by $\Phi_t^{*h}$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 2.1
  • Proposition 2.2: Pointing Towards $\mathsf{Ein}^{p-1,p}$
  • proof
  • Proposition 2.3: Fiber Bundle for $\mathsf{Ein}^{p-1,p}$
  • proof
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 31 more