Abstract. We study the isometry group of a globally hyperbolic spatially compact Lorentz surface. Such a group acts on the circle, and we show that when the isometry group acts non properly, the subgroups of Diff(S 1 ) obtained are semi conjugate to subgroups of finite covers of PSL(2, R) by using convergence groups. Under an assumption on the conformal boundary, we show that we have a conjugacy in Homeo(S 1 ).
The aim of this article is to understand the geometry of limit sets in pseudo-Riemannian hyperbolic geometry. We focus on a class of subgroups of PO(p, q + 1) introduced by Danciger, Guéritaud and Kassel, called H p,q -convex cocompact. We define a pseudo-Riemannian analogue of critical exponent and Hausdorff dimension of the limit set. We show that they are equal and bounded from above by the usual Hausdorff dimension of the limit set. We also prove a rigidity result in H 2,1 = AdS 3 which can be understood as a Lorentzian version of a famous Theorem of R. Bowen in 3-dimensional hyperbolic geometry.
Limit sets of AdS-quasi-Fuchsian groups of PO(n, 2) are always Lipschitz submanifolds. The aim of this article is to show that they are never C 1 , except for the case of Fuchsian groups. As a byproduct we show that AdS-quasi-Fuchsian groups that are not Fuchsian are Zariski dense in PO(n, 2).
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