# Affine actions with Hitchin linear part

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## 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 \({{\mathsf {S}}}{{\mathsf {O}}}(n,n-1)\), 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.

## Notes

### Acknowledgements

We thank Bill Goldman, Oliver Guichard, Fanny Kassel, and Qiongling Li for valuable discussions. We also acknowledge the independent work of Sourav Ghosh [Gho18], already mentioned in Remark 1.4, which includes similar statements to our Lemma 8.2 and Theorem 8.7 (see Proposition 0.0.1 and Theorem 0.0.2), and also an alternate proof of Theorem 6.1 based on ideas of [GT17] (see Theorem 0.0.3). When we recently found out about this work, we discussed our results with Ghosh. Thanks to those discussions, we realized that the techniques from Sections 7 and 8, which are similar to Ghosh’s techniques, and which we had originally written in the more narrow context of affine actions whose linear part is Anosov with respect to the minimal parabolic, actually apply in the more general context where the linear part is only assumed to be Anosov with respect to the stabilizer of an isotropic \((n-1)\)-plane, as in Theorem 8.7. We also thank the Mathematisches Forschungsinstitut Oberwolfach for hosting both authors in September 2018 for the meeting “New trends in Teichmüller theory and mapping class groups", at which the first author gave a lecture about this project. See [DZ18] for a short description of the contents of the lecture. The first author acknowledges with regret that at the time of that lecture, he was not aware that Theorem 7.10 had already been shown by Ghosh–Treib [GT17] in full generality, and he failed to cite that work during the lecture.

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