AFFINE ACTIONS WITH HITCHIN LINEAR PART

Abstract

Properly discontinuous actions of a surface group by affine automorphisms of Rd 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 Rd 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(2 n, 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.