Anosov structures on Margulis spacetimes

Abstract

In this paper we describe the stable and unstable leaves for the geodesic flow on the space of non-wandering geodesics of a Margulis Space Time and prove contraction properties of the leaves under the flow. We also show that mondromy of Margulis Space Times are "Anosov representations in non semi-simple Lie groups".

Affine actions with Hitchin linear part

“…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.…”

“…The Margulis invariant for currents and the properness criterion. Here we will discuss a properness criterion for actions on E n,n´1 , due originally to Goldman-Labourie-Margulis [GLM09] in the case of free and surface groups acting on E 2,1 , and extended by Ghosh-Trieb [GT17] to the case of word hyperbolic groups acting with Anosov linear part in any E n,n´1 . This is one of several key tools needed for Theorem 1.2.…”

“…Now, since ι n,n˝ρ has trivial centralizer in PSOpn, nq, results of Goldman [Gol84] on representation varieties of surface groups imply that any ρ-cocycle u : π 1 S Ñ R n,n´1 is realized as the R n,n´1 part of the derivative of a smooth deformation path ε as above (and the sopn, n´1q part may be taken to be trivial). We prove a key lemma (Lemma 8.2) that connects a criterion [GLM09,GT17] for properness of the affine action pρ, uq on E n,n´1 with the first order behavior of the two middle eigenvalues of elements ε pγq, an inverse pair λ n , λ´1 n which converges to the two one-eigenvalues of ι n,n˝ρ as ε Ñ 0. From this eigenvalue behavior, we use [GGKW17, KLP14,KLP15] to prove that if pρ, uq acts properly on R n,n´1 , then the representations ε satisfy an unexpected Anosov condition.…”

“…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-Trieb [GT17] in full generality, and he failed to cite that work during the lecture.…”

An introduction to pressure metrics for higher Teichmüller spaces

“…They were originally discovered by Margulis [59] as counterexamples to a question of Milnor. Ghosh [32] used work of Goldman, Labourie and Margulis [34,35] to interpret holonomy maps of Margulis space times (without cusps) as "Anosov representations" into the (non-semisimple) Lie group Aff(R 3 ) of affine automorphisms of R 3 . Ghosh [33] was then able to adapt the techniques of [17] to produce a pressure form on the analytic manifold M of (conjugacy classes of) holonomy maps of Margulis space times of fixed rank (with no cusps).…”