Avatars of Margulis invariants and proper actions

Abstract

In this article, we interpret affine Anosov representations of any word hyperbolic group in SO0(n − 1, n) ⋉ R 2n−1 as infinitesimal versions of representations of word hyperbolic groups in SO0(n, n) which are both Anosov in SO0(n, n) with respect to the stabilizer of an oriented (n−1)-dimensional isotropic plane and Anosov in SL(2n, R) with respect to the stabilizer of an oriented n-dimensional plane. Moreover, we show that representations of word hyperbolic groups in SO0(n, n) which are Anosov in SO0(n, n) with respect to the stabilizer of an oriented (n − 1)dimensional isotropic plane, are Anosov in SL(2n, R) with respect to the stabilizer of an oriented n-dimensional plane if and only if its action on SO0(n, n)/SO0(n − 1, n) is proper. In the process, we also provide various different interpretations of the Margulis invariant.

Affine Anosov Representations and Proper Actions

“…Similar results relating Margulis invariants and Jordan projections has also been independently obtained by Andrés Sambarino and Kassel-Smilga [KSon]. Furthermore, we use the stability of Anosov representations under small deformations to prove the following existence result (similar result but in a different setting was previously obtained by Ghosh [Gho18]):…”

“…We denote the affine cross ratio of the four mutually transverse affine parabolic subspaces by β(A + , A − , (g, Y )A, A) (see Definition 4.1.4 for more details). We generalize results from Ghosh [Gho18] and prove that:…”

“…Proof. We use Theorem 4.2.2 and follow verbatim the proof of Propositions 2.3.4 and 2.3.5 of Ghosh [Gho18]. □…”