The conformal boundary of Margulis space--times.
Let \(\mathbb R^{1,2}/\Gamma\) be a Margulis space-time where \(\Gamma\) is an affine Schottky group. The group \(\Gamma\) acts on the Einstein universe \textbf{Ein}\(_3\) as a discrete group of conformal transformations. This action is properly discontinuous on a subset conformally equivalent to \(\mathbb R^{1,2}\), and a natural question is to know if the action remains properly discontinuous on a greater open set, intersecting the boundary \(\mathbb C_\infty\). The author answers this question in an affirmative way in the case where \(\Gamma\) is Drumm's affine Schottky group. The main result of the paper is stated as follows: For each affine Schottky group \(\Gamma\subset\text{Is}(\mathbb R^{1,2})\), the Margulis space-time \(M_\Gamma=\mathbb R^{1,2}/\Gamma\) admits a conformal boundary \(\partial M_\Gamma\). More precisely, there is a closed subset \(\Lambda_\Gamma^\infty\) of \(\mathbb C_\infty\), such that: (i) The action of \(\Gamma\) is free and properly discontinuous on \(\Omega_\Gamma=\)\textbf{Ein}\(_3\setminus\Lambda_\Gamma^\infty\), and \(\Omega_\Gamma\) is the biggest open subset of \textbf{Ein}\(_3\) containing \(\mathbb R^{1,2}\), having this property. (ii) The quotient space \((\Omega_\Gamma^\infty\cup\mathbb R^{1,2})/\Gamma\) (where \(\Omega_\Gamma^\infty =\mathbb C_\infty\backslash\Lambda_\Gamma^\infty)\) is a smooth manifold with boundary, whose interior is conformally equivalent to the Margulis space-time \(M_\Gamma\). The boundary \(\Omega_\Gamma^\infty/\Gamma\) is a finite union of cylinders \(\mathbb R\times\mathbb S^1\), each endowed with a conformal class of degenerate metrics. (iii) There is a two points compactification of \(\Omega_\Gamma^\infty/\Gamma\) denoted by \(\partial M_\Gamma\). The space \(M_\Gamma\cup\partial M_\Gamma\) is singular in these two points. Each semi-geodesic of \(M_\Gamma\) leaving any compact subset of \(M_\Gamma\), tends to a unique point of \(\partial M_\Gamma\).
discrete group of conformal transformations
conformal boundary of a Lorentzian manifold
scientific article; zbMATH DE number 1981359