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.


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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