Affine actions with Hitchin linear part

Abstract

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 SOpn, n´1q, so that the affine action is by isometries of a flat pseudo-Riemannian metric on R d of signature pn, n´1q. Here, the translational part determines a deformation of the linear part into PSOpn, nq-Hitchin representations and the crucial step is to show that such representations are not Anosov in PSLp2n, Rq 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 pn, n´1q by a PSOpn, nq-Hitchin representation fails to be properly discontinuous.

Avatars of Margulis invariants and proper actions

“…Lastly, we would like to mention that Danciger-Zhang has announced independent work in [DZ18] which has overlap with some of our results. In particular, Proposition 0.0.1, Theorem 0.0.2 and Theorem 0.0.3 of this article when applied to fundamental groups of compact surfaces without boundary, are respectively similar to Lemma 8.2, Theorem 8.8 and Theorem 6.1 of [DZ18]. On the other hand, the results about crossratios contained in Sections 2.3 and 2.4 are not obtained by Danciger-Zhang.…”

“…On the other hand, the results about crossratios contained in Sections 2.3 and 2.4 are not obtained by Danciger-Zhang. We would also like to note that, even though Theorem 0.0.4 has not been stated as a result in [DZ18], for the case of fundamental groups of compact surfaces without boundary, it can also be obtained by jointly applying Theorems 8.8 and 6.1 of [DZ18]. In [DZ18], Danciger-Zhang use Lemma 8.2, Theorem 8.8, Theorem 6.1 and they also use certain properties special to Hitchin representations obtained from the works of Labourie [Lab06] and Fock-Goncharov [FG06] to generalize Theorem 1.1 of [Lab01] and conclude that representations in PSL(2n − 1, R) ⋉ R 2n−1 whose linear parts are Hitchin do not admit proper affine actions on R 2n−1 .…”

Topological restrictions on Anosov representations

“…Note that Danciger and Zhang [22, Theorem 1.3] proved that Hitchin representations into are not ‐Anosov, if you regard them as representations into .…”

Conformality for a robust class of non-conformal attractors

“…Furthermore it is easy to check that the n-th exterior power ^n : PSOpn, nq Ñ PSLp^nR 2n q splits as the direct sum of two irreducible PSOpn, nq-modules, which have respectively a n´1 and b n as first root (see for example Danciger-Zhang [19]). In particular we obtain the following result, independently announced by Labourie [36].…”

“…Remark 9.10. Danciger-Zhang [19] recently proved that when a representation ρ P H pS, PSOpn, nqq is regarded as a representation in PSL 2n pRq, it is, instead, never ta n u-Anosov and the limit curve in the n ´1-Grassmannian is never C 1 . and define f λ1 ρ : Λ Γ Ñ R by f λ1 ρ ptg i u iPZ q " log }ρpg 0 qv} }v} for any v P px tgiu q 1 ρ .…”