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Geometric structures for maximal representations and pencils

Colin Davalo

Abstract

We study fibrations of the projective model for the symmetric space associated with $\text{SL}(2n,\mathbb{R})$ by codimension $2$ projective subspaces, or pencils of quadrics. In particular we show that if such a smooth fibration is equivariant with respect to a representation of a closed surface group, the representation is quasi-isometrically embedded, and even Anosov if the pencils in the image contain only non-degenerate quadrics. We use this to characterize maximal representations among representations of a closed surface group into $\text{Sp}(2n,\mathbb{R})$ by the existence of an equivariant continuous fibration of the associated symmetric space, satisfying an additional technical property. These fibrations extend to fibrations of the projective structures associated to maximal representations by bases of pencils of quadrics.

Geometric structures for maximal representations and pencils

Abstract

We study fibrations of the projective model for the symmetric space associated with by codimension projective subspaces, or pencils of quadrics. In particular we show that if such a smooth fibration is equivariant with respect to a representation of a closed surface group, the representation is quasi-isometrically embedded, and even Anosov if the pencils in the image contain only non-degenerate quadrics. We use this to characterize maximal representations among representations of a closed surface group into by the existence of an equivariant continuous fibration of the associated symmetric space, satisfying an additional technical property. These fibrations extend to fibrations of the projective structures associated to maximal representations by bases of pencils of quadrics.
Paper Structure (29 sections, 45 theorems, 97 equations, 7 figures)

This paper contains 29 sections, 45 theorems, 97 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Fibrations of H3 by geodesics.
  3. Fibrations of a convex set.
  4. Maximal representations in Sp(2n,R).
  5. Characterization of maximal representations.
  6. Organization of the paper
  7. Maximal and Anosov representations.
  8. Maximal representations.
  9. Anosov representations.
  10. An invariant convex domain and its fibrations
  11. Pencils of quadrics
  12. Fitting pairs.
  13. Fitting directions.
  14. Fibration of a convex set and globally fitting maps.
  15. Fitting flows.
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $\rho:\Gamma_g\to \mathop{\mathrm{SL}}\nolimits(2,\mathbb{C})$ be a representation that admits an equivariant fitting immersion $u:\widetilde{S_g}\to \mathcal{G}$. The representation $\rho$ is quasi-Fuchsian.

Figures (7)

  • Figure 1: Illustration of Proposition \ref{['prop:TriangIneqCr']}
  • Figure 2: Illustration of Proposition \ref{['prop:Intersection linear']}
  • Figure 3: Illustration of the proof of Theorem \ref{['thm:WellFittedImplesAnosov']}.
  • Figure 4: Proof of Lemma \ref{['prop:Exist fitting flow']}.
  • Figure 5: Three Hermitian quadrics in a pencil and the corresponding geodesic in $\mathbb{H}^3$.
  • ...and 2 more figures

Theorems & Definitions (101)

  • Theorem 1.1
  • Theorem 1.2: Theorem \ref{['thm:Quasi-Fuchsian with no fitting']}
  • Theorem 1.3
  • Theorem 1.5: Theorem \ref{['thm: fitting implies maximal']}
  • Theorem 1.6: Theorem \ref{['thm: fitting implies maximal']}
  • Corollary 1.7
  • Lemma 2.1: BIW10
  • Definition 2.2: BPS
  • Theorem 2.3: Gu_ritaud_2017
  • Theorem 2.4
  • ...and 91 more