Geometry and topology of complete Lorentz spacetimes of constant curvature (Q2801753)

Complete Lorentzian manifolds which are quotients of the anti-de Sitter spacetime \(\mathrm{AdS}^3\) or Minkowski spacetime \(\mathbb R^{2,1}\) by discrete groups \(\Gamma\) acting properly discontinuously by isometries are considered. There are the so-called standard examples of such actions and corresponding quotients, deformations of these standard ones and also other examples, known from previous works. In particular, there exist nonsolvable groups and nonabelian free groups acting properly discontinuously on \(\mathbb R^{2,1}\). The quotient manifolds from the actions of the latter are called Margulis spacetimes.NEWLINENEWLINENEWLINEA new criterion for the properly discontinuous action of a convex cocompact representation of a discrete group \(\Gamma\) is formulated in the paper using the infinitesimal Lipschitz contraction and infinitesimal length contraction. The geometric and topological descriptions of flat Lorentzian manifolds using the infinitesimal Lipschitz contraction criterion are derived. The positive answer to the conjecture that Margulis spacetimes should be topologically tame (homeomorphic to the interior of a compact manifold) is obtained in both the flat and the negatively curved cases. In the compact case, is was known from previous works. The present result is the first contribution to the topology of noncompact quotients of \(\mathrm{AdS}^3\).NEWLINENEWLINENEWLINEThe fact that \(\mathbb R^{2,1}\) is the tangent space of \(\mathrm{AdS}^3\), allows to consider Lorentzian manifolds modeled on \(\mathbb R^{2,1}\) as infinitesimal versions of Lorentzian manifolds modeled on \(\mathrm{AdS}^3\). A geometric transition from \(\mathrm{AdS}^3\) geometry to \(\mathbb R^{2,1}\) geometry is developed in the paper. It allows to find collapsing \(\mathrm{AdS}^3\) manifolds which limit to a given Margulis spacetime.