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Proper affine actions and geodesic flows of hyperbolic surfaces. (English) Zbl 1193.57001

In this paper, \(\Gamma_0\subset \mathrm{O}(2,1)\) is a Schottky group, \(\Sigma=\mathbb{H}^2/\Gamma_0\) is the corresponding hyperbolic surface and \(\mathcal{C}(\Sigma)\) denotes the space of unit length geodesic currents on \(\Sigma\). An affine deformation of \(\Gamma_0\) is a group \(\Gamma\) of affine transformations whose linear part \(\mathbb{L}\) equals \(\Gamma_0\), that is, a subgroup \(\Gamma\subset \mathrm{Isom}^0(\mathbb{E}^{2,1})\) such that the restriction of \(\mathbb{L}\) to \(\Gamma\) is an isomotphism \(\Gamma\to \Gamma_0\). The cohomology group \(H^1(\Gamma_0,V)\) parametrizes equivalence classes of affine deformations \(\Gamma_u\subset \mathrm{O}(2,1)^0\) of \(\Gamma_0\) acting an an irreducible holonomy representation \(V\) of \(\mathrm{O}(2,1)\). Here, \(u\) is a cocycle in \(Z^1(\Gamma_0,V)\).
The authors define a continuous biaffine map \(\Psi:\mathcal{C}(\Sigma)\times H^1(\Gamma_0,V)\to \mathbb{R}\) which is linear on the vector space \(H^1(\Gamma_0,V)\). They show that an affine deformation \(\Gamma_u\) acts properly if and only if \(\Psi(\mu,[u])\not=0\) for all \(\mu\in \mathcal{C}(\Sigma)\). As a result, the set of proper affine actions whose linear part is a Schottky group is identified with a bundle of open convex cones in \(H^1(\Gamma_0,V)\) over the Fricke-Teichmüller space of \(\Sigma\).
An essential tool used in this work is a generalization of an invariant for affine deformations constructued by G. Margulis in [Sov. Math., Dokl. 28, 435–439 (1983); translation from Dokl. Akad. Nauk SSSR 272, 785–788 (1983; Zbl 0578.57012)] and extended to higher dimenstions by F. Labourie in [J. Differ. Geom. 59, No. 1, 15–31 (2001; Zbl 1037.57031)].

MSC:

57M05 Fundamental group, presentations, free differential calculus
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
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References:

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