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.

中文翻译：

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Hitchin线性零件的仿射动作

Danciger–Gueritaud–Kassel已证明存在基于\（{\ mathbb {R}} ^ d \的仿射自同构的表面组的正确不连续动作。但是，我们表明，如果仿射表面组动作的线性部分位于Hitchin组件中，则该动作将无法正确地中断。关键情况是\（{{\ mathsf {S}}} {{\ mathsf {O}}}（n，n-1）\）中的线性部分，因此仿射作用是通过平面的等距签名\（（n，n-1）\）\\ {{\ mathbb {R}} ^ d \）上的伪黎曼度量。在此，平移部分将线性部分的变形确定为\（\ mathsf {PSO}（n，n）\）- Hitchin表示，关键的一步是证明这种表示不是Anosov in关于n平面的稳定器，为\（\ mathsf {PSL}（2n，{\ mathbb {R}}）\）。我们还证明了主要结果的负曲率类似物，即表面群对签名\（（n，n-1）\）的伪黎曼双曲空间的作用是\（\ mathsf {PSO}（n ，n）\）-希金表示法不能正确地间断。