Geometric structures for maximal representations and pencils

TL;DR

This paper investigates geometric fibrations related to maximal representations of surface groups, showing how certain equivariant fibrations imply quasi-isometric embedding and Anosov properties, and characterizing maximal representations via these fibrations.

Contribution

It introduces a new geometric framework using pencils of quadrics to characterize maximal representations and their properties in symplectic groups.

Findings

Equivariant fibrations imply quasi-isometric embeddings.

Non-degenerate quadrics in pencils lead to Anosov representations.

Characterization of maximal representations via fibrations of symmetric spaces.

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.