Volume 5, issue 1 (2001)

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Flag structures on Seifert manifolds

Thierry Barbot

Geometry & Topology 5 (2001) 227–266

arXiv: math.DS/0104108


We consider faithful projective actions of a cocompact lattice of SL(2, ) 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(2, ), viewed as the group of linear transformations of a copy of the affine plane in P2. 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.

flag structure, transverserly affine structure
Mathematical Subject Classification 2000
Primary: 57R50, 57R30
Secondary: 32G07, 58H15
Forward citations
Received: 23 January 1999
Revised: 3 April 2000
Accepted: 19 March 2001
Published: 23 March 2001
Proposed: Jean-Pierre Otal
Seconded: Yasha Eliashberg, David Gabai
Thierry Barbot
ENS Lyon
UMR 5669
46 Allée d’Italie
69364 Lyon