Flag structures on Seifert manifolds

Abstract

We consider faithful projective actions of a cocompact lattice of $SL\left(2,ℝ\right)$ on the projective plane, with the following property: there is a common fixed point, which is a saddle fixed point for every element of infinite order of the the group. Typical examples of such an action are linear actions, ie, when the action arises from a morphism of the group into $GL\left(2,ℝ\right)$, viewed as the group of linear transformations of a copy of the affine plane in $ℝP^2$. We prove that in the general situation, such an action is always topologically linearisable, and that the linearisation is Lipschitz if and only if it is projective. This result is obtained through the study of a certain family of flag structures on Seifert manifolds. As a corollary, we deduce some dynamical properties of the transversely affine flows obtained by deformations of horocyclic flows. In particular, these flows are not minimal.