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Properly discontinuous actions of a surface group by affine automorphisms of 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 𝖲𝖮(n,n−1), so that the affine action is by isometries of a flat pseudo-Riemannian metric on ℝ^d of signature (n,n−1). Here, the translational part determines a deformation of the linear part into 𝖯𝖲𝖮(n,n)-Hitchin representations and the crucial step is to show that such representations are not Anosov in 𝖯𝖲𝖫(2n,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 𝖯𝖲𝖮(n,n)-Hitchin representation fails to be properly discontinuous.more » « less
