GHMC manifold

GHMC manifolds ( global hyperbolic maximally Cauchy compact manifolds ) are a geometrical concept that is used as a simple model for (2 + 1) -dimensional gravity .

definition

A Lorentz manifold is said to be globally hyperbolic if “traveling forward in time never leads to the past”, i. H. if there is a space-like hypersurface ( Cauchy surface ) that meets every maximum causal curve exactly once.

A d-dimensional Lorentz manifold is called a globally hyperbolic maximally Cauchy-compact manifold if it is globally hyperbolic, the Cauchy surface is compact and with these properties is maximal, i.e. H. cannot be embedded isometrically in a larger global hyperbolic d-dimensional Lorentz manifold. ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$

Set of mess

Mess's theorem (after Geoffrey Mess 1990) describes the 3-dimensional GHMC manifolds that are diffeomorphic to for a surface . ${\displaystyle S\times \left(0,1\right)}$ ${\displaystyle S}$

His description uses the identification of the 3-dimensional anti-de-sitter space with and its isometric group with , under which the effect of on corresponds to the effect of on through left and right multiplication. ${\displaystyle AdS^{3}}$ ${\displaystyle PSL(2,\mathbb {R} )}$ ${\displaystyle PO_{0}(2,2)}$ ${\displaystyle SL(2,\mathbb {R} )\times SL(2,\mathbb {R} )}$ ${\displaystyle PO_{0}(2,2)}$ ${\displaystyle AdS^{3}}$ ${\displaystyle SL(2,\mathbb {R} )\times SL(2,\mathbb {R} )}$ ${\displaystyle PSL(2,\mathbb {R} )}$

With this identification, each GHMC structure is obtained as follows. Be a pair of Fuchsian groups . The effect of on leaves a circle invariant, namely the graph of the effect of in the effect of conjugating homeomorphism . Let be the complement of the union of all (with regard to the Lorentz metric) for . Then ${\displaystyle S\times \left(0,1\right)}$ ${\displaystyle (\rho _{L},\rho _{R})(\pi _{1}S)}$ ${\displaystyle (\rho _{L},\rho _{R})(\pi _{1}S)}$ ${\displaystyle \partial _{\infty }AdS^{3}}$ ${\displaystyle \Lambda }$ ${\displaystyle \rho _{L}}$ ${\displaystyle \rho _{R}}$ ${\displaystyle \mathbb {R} P^{1}\to \mathbb {R} P^{1}}$ ${\displaystyle \Omega \subset \mathbb {R} P^{3}}$ ${\displaystyle z^{\perp }}$ ${\displaystyle z\in \Lambda }$

${\displaystyle (\rho _{L},\rho _{R})(\pi _{1}S)\backslash \Omega }$

a GHMC manifold becomes diffeomorphic, and Mess's theorem states that every GHMC structure arises in this way. The holonomy groups of these GHMC structures are also known as AdS quasi-fox groups . ${\displaystyle S\times \left(0,1\right)}$ ${\displaystyle S\times \left(0,1\right)}$

literature

• Thierry Barbot: Lorentzian Kleinian Groups . In: L. Ji, A. Papadopoulos, S.-T. Yau (Ed.): Handbook of group actions , Volume 1. International Press and Higher Education Press, 2015, arxiv : 1609.03863
• Lars Andersson, Thierry Barbot, Riccardo Benedetti, Francesco Bonsante, William M. Goldman, François Labourie, Kevin Scannell, Jean-Marc Schlenker: Notes on a Paper of Mess . In: Geometriae Dedicata , Volume 126, 2007, pp. 47-70, arxiv : 0706.0640

Individual evidence

1. ^ Edward Witten : 2 + 1 dimensional gravity as an exactly soluble system . In: Nuclear Physics , B311, 46-78 (1988). fis.puc.cl
2. ↑ Geoffrey Mess: Lorentz spacetimes of constant curvature . MSRI Preprint 1990. In: Geometriae Dedicata , Volume 126, 2007, pp. 3-45, arxiv : 0706.1570
3. ^ Fanny Kassel : Geometric structures and representations of discrete groups . Proceedings of the International Congress of Mathematicians, Rio de Janeiro 2018. arxiv : 1802.07221