Properness criteria for affine actions of Anosov groups

I will present some criteria (necessary or sufficient) for the action on the affine space of a group Gamma of affine transformations to be proper. This is joint work with Fanny Kassel.

The main of these criteria links properness of action to the divergence of a parameter called the Margulis invariant. This invariant measures roughly the translation part of an affine transformation, but in a way that is invariant by conjugation.

This link was already known in some special cases (and has often been exploited to construct proper actions). We tried to establish it in as general setting as possible. We proved it in particular if Gamma has some suitable Anosov property (with respect to some natural parabolic subgroup, that depends on the affine group we are working in).

I will possibly also evoke some other invariants similar to the Margulis invariant, that could lead to criteria that work in even more general settings.