Geometric structures for maximal representations and pencils

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.

💡 Research Summary

The paper investigates fibrations of the projective model of the symmetric space associated with the Lie group SL(2n,ℝ) by codimension‑2 projective subspaces, which the author calls “pencils of quadrics.” A pencil is a two‑dimensional linear family of quadratic forms on ℝ^{2n}, i.e. a plane in the dual space Q = S²(ℝ^{2n})*. The projective convex domain ℙ(S²V⁺) (the projectivization of the cone of positive definite symmetric tensors) serves as the model for the symmetric space X = SL(2n,ℝ)/SO(2n).

The first main theme is to understand when a representation ρ of a closed surface group Γ_g into SL(2n,ℝ) admits a ρ‑equivariant “fitting immersion” u : ˜S_g → Gr_{mix}²(Q), where Gr_{mix}²(Q) denotes the set of pencils that intersect the convex domain non‑trivially (i.e. pencils whose members have mixed signature and contain no non‑zero semi‑positive quadric). A fitting immersion is defined by the property that the associated codimension‑2 projective subspaces, together with a choice of co‑orientation, give rise to a family of half‑spaces in ℙ(S²V⁺) that vary continuously and are nested along the leaves of a flow.

The author introduces the notion of a “fitting flow.” For each point x∈˜S_g the immersion u determines a circle bundle of co‑oriented hyperplanes containing the codimension‑2 subspace associated to u(x). A fitting flow is a smooth flow on this circle bundle such that, as one moves along a flow line, the corresponding half‑spaces in ℙ(S²V⁺) become strictly smaller. This nesting property yields uniform contraction in the projective space, which is precisely the dynamical condition used in the Bochi‑Potrie‑Sambarino criterion for Anosov representations (nested multicones).

Using this machinery, the paper proves:

• Theorem 1.3. If ρ : Γ_g → SL(2n,ℝ) admits a ρ‑equivariant fitting immersion whose image lies in the subset Gr_{p,n; p,n}(Q) of pencils consisting only of non‑degenerate quadrics (i.e. each non‑zero element has signature (p,n) with p=n), then ρ is t‑Anosov for every t∈(0,n). In particular, the representation is Anosov in the sense of Labourie–Guichard–Wienhard.

• Theorem 5.4 (general version). The existence of a continuous ρ‑equivariant map into a suitable Grassmannian together with a fitting flow already forces the representation to be Anosov, even when the map is only continuous and not smooth.

The second major part of the paper focuses on the symplectic group Sp(2n,ℝ). The symmetric space X_{Sp}=Sp(2n,ℝ)/U(n) embeds as a totally geodesic submanifold of X_{SL}. The Lagrangian Grassmannian L_n (the space of n‑dimensional isotropic subspaces) plays a central role. The author defines a distinguished open subset Gr_ω²(Q)⊂Gr₂(Q) consisting of pencils that contain a quadratic form positive on some Lagrangian and negative on another. This set has two connected components; one of them is denoted Gr_{max}²(Q) and is singled out for the characterization of maximal representations.

The main results concerning Sp(2n,ℝ) are:

• Theorem 1.5. If a representation ρ : Γ_g → Sp(2n,ℝ) admits a ρ‑equivariant fitting immersion u : ˜S_g → Gr_{max}²(Q), then ρ is maximal, i.e. its Toledo invariant attains the maximal possible value (2g‑2)n. Moreover, any fitting immersion whose image lies in Gr_{p,n; p,n}(Q) must actually land inside Gr_ω²(Q).

• Theorem 1.6. Conversely, if ρ is maximal then there exists a continuous ρ‑equivariant map u : ˜S_g → Gr_{max}²(Q) which admits a fitting flow. Although u need not be smooth, the existence of the flow guarantees a continuous fibration of the domain of discontinuity in projective space constructed by Guichard–Wienhard. The quotient inherits a natural (Sp(2n,ℝ), ℝP^{2n‑1})‑contact structure.

Thus maximal representations are exactly those surface group representations into Sp(2n,ℝ) that admit a continuous equivariant fibration of the associated locally symmetric space by pencils of quadrics satisfying the fitting‑flow condition. This provides a new geometric characterization parallel to the classical description via boundary maps into the Lagrangian Grassmannian.

The paper also contains several auxiliary results:

• In Section 1.1 the author revisits the classical picture of H³ fibrated by geodesics, showing how a quasi‑Fuchsian representation admits a fitting immersion given by the Gauss map of an equivariant surface with principal curvature in (−1,1). This serves as a motivating example.

• Appendix A constructs quasi‑Fuchsian representations that admit no fitting immersion, demonstrating that the existence of a fitting immersion is a strictly stronger condition than being quasi‑Fuchsian.

• Appendix B analyses the case Sp(4,ℝ) in detail, relating spacelike surfaces in the associated pseudo‑Riemannian symmetric space to fitting immersions of pencils.

Overall, the paper introduces the novel concept of fitting flows to translate geometric data (pencils of quadrics) into dynamical contraction properties, thereby linking the existence of equivariant fibrations to Anosov and maximal representation theory. This bridges higher‑rank Teichmüller theory, the geometry of symmetric spaces, and the dynamics of surface group actions on flag manifolds, offering a fresh perspective on the structure of maximal representations.