# Affine actions with Hitchin linear part

## TL;DR

The paper demonstrates that surface group actions with Hitchin linear parts on affine and pseudo-Riemannian spaces are not properly discontinuous, contrasting with previous results for other affine actions.

## Contribution

It shows that Hitchin component linear parts cannot produce properly discontinuous affine actions, revealing limitations of Hitchin representations in geometric group actions.

## Key findings

- Hitchin linear parts lead to non-proper affine actions.
- Hitchin representations in $ ext{PSO}(n,n)$ are not Anosov in $ ext{PSL}(2n, ext{R})$.
- Pseudo-Riemannian hyperbolic space actions are not properly discontinuous with Hitchin representations.

## Abstract

Properly discontinuous actions of a surface group by affine automorphisms of $\mathbb R^d$ were shown to exist by Danciger-Gueritaud-Kassel. We show, however, that if the linear part of an affine surface group action is in the Hitchin component, then the action fails to be properly discontinuous. The key case is that of linear part in $\mathsf{SO}(n,n-1)$, so that the affine action is by isometries of a flat pseudo-Riemannian metric on $\mathbb R^d$ of signature $(n,n-1)$. Here, the translational part determines a deformation of the linear part into $\mathsf{PSO}(n,n)$-Hitchin representations and the crucial step is to show that such representations are not Anosov in $\mathsf{PSL}(2n,\mathbb R)$ with respect to the stabilizer of an $n$-plane. We also prove a negative curvature analogue of the main result, that the action of a surface group on the pseudo-Riemannian hyperbolic space of signature $(n,n-1)$ by a $\mathsf{PSO}(n,n)$-Hitchin representation fails to be properly discontinuous.