#### Geometry & Topology, Vol. 5 (2001)
Paper no. 7, pages 227--266.

## Flag Structures on Seifert Manifolds

### Thierry Barbot

**Abstract**.
We consider faithful projective actions of a cocompact lattice of
SL(2,R) 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,R), 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.
**Keywords**.
Flag structure, transverserly affine structure

**AMS subject classification**.
Primary: 57R50, 57R30.
Secondary: 32G07, 58H15.

**DOI:** 10.2140/gt.2001.5.227

**E-print:** `arXiv:math.DS/0104108`

Submitted to GT on 23 January 1999.
(Revised 3 April 2000.)
Paper accepted 19 March 2001.
Paper published 23 March 2001.

Notes on file formats
Thierry Barbot

C.N.R.S, ENS Lyon, UMPA, UMR 5669

46, allee d'Italie, 69364 Lyon, France

Email: barbot@umpa.ens-lyon.fr

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