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Isometries of Lorentz surfaces and convergence groups

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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 \(\mathrm {Diff}(\mathbb {S}^1)\) obtained are semi conjugate to subgroups of finite covers of \(\mathrm {PSL}(2,\mathbb {R})\) by using convergence groups. Under an assumption on the conformal boundary, we show that we have a conjugacy in \(\mathrm {Homeo}(\mathbb {S}^1)\).

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Notes

  1. A Riemannian manifold on which the conformal group acts non properly is conformally diffeomorphic to the round sphere or to the Euclidian space.

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Acknowledgments

This work corresponds to chapters 1 and 3 of my PhD thesis [30]. I would like to thank my advisor Abdelghani Zeghib for his help throughout this work, as well as Thierry Barbot for some important remarks on a first version of this paper. This work was partially supported by ANR project GR-Analysis-Geometry (ANR-2011-BS01-003-02).

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  1. UMPA, École Normale Supérieure de Lyon, 46 allée d’Italie, 69364, Lyon Cedex 07, France

    Daniel Monclair

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Monclair, D. Isometries of Lorentz surfaces and convergence groups. Math. Ann. 363, 101–141 (2015). https://doi.org/10.1007/s00208-014-1157-9

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