## Geometry & Topology

- Geom. Topol.
- Volume 5, Number 1 (2001), 227-266.

### Flag structures on Seifert manifolds

#### Abstract

We consider faithful projective actions of a cocompact lattice of $SL\left(2,\mathbb{R}\right)$ 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 $GL\left(2,\mathbb{R}\right)$, viewed as the group of linear transformations of a copy of the affine plane in $\mathbb{R}{P}^{2}$. 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.

#### Article information

**Source**

Geom. Topol., Volume 5, Number 1 (2001), 227-266.

**Dates**

Received: 23 January 1999

Revised: 3 April 2000

Accepted: 19 March 2001

First available in Project Euclid: 21 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.gt/1513882988

**Digital Object Identifier**

doi:10.2140/gt.2001.5.227

**Mathematical Reviews number (MathSciNet)**

MR1825662

**Zentralblatt MATH identifier**

1032.57037

**Subjects**

Primary: 57R50: Diffeomorphisms 57R30: Foliations; geometric theory

Secondary: 32G07: Deformations of special (e.g. CR) structures 58H15: Deformations of structures [See also 32Gxx, 58J10]

**Keywords**

flag structure transverserly affine structure

#### Citation

Barbot, Thierry. Flag structures on Seifert manifolds. Geom. Topol. 5 (2001), no. 1, 227--266. doi:10.2140/gt.2001.5.227. https://projecteuclid.org/euclid.gt/1513882988