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Causal properties of AdS-isometry groups I: Causal actions and limit sets. (English) Zbl 1151.83311
Author’s abstract: We study the causality relation in the 3-dimensional anti-de Sitter space AdS and its conformal boundary Ein$$_2$$. To any closed achronal subset $$\Lambda$$ in Ein$$_2$$ we associate the invisible domain $$E(\Lambda)$$ from $$\Lambda$$ in AdS. We show that if $$\Gamma$$ is a torsion-free discrete group of isometries of AdS preserving $$\Lambda$$ and is non-elementary (for example, not abelian) then the action of $$\Gamma$$ on $$E(\Lambda)$$ is free, properly discontinuous and strongly causal. If $$\Lambda$$ is a topological circle then the quotient space $$M_\Lambda(\Gamma) = \Gamma\setminus E(\Lambda)$$ is a maximal globally hyperbolic AdS-spacetime admitting a Cauchy surface $$S$$ such that the induced metric on $$S$$ is complete. In a forthcoming paper we study the case where $$\Gamma$$ is elementary and use the results of the present paper to define a large family of AdS-spacetimes including all the previously known examples of BTZ multi-black holes.

##### MSC:
 83C15 Exact solutions to problems in general relativity and gravitational theory 83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory 83F05 Relativistic cosmology
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