Heintze Group
In geometry , Heintze groups are certain solvable Lie groups which, according to Ernst Heintze's theorem, are the only negatively curved homogeneous spaces .
They are important as an important class of examples in Large Scale Geometry. An open conjecture says that Heintze groups are only quasi-isometric if they are isomorphic.
Definitions
A Heintze group is a Lie group, which is a semidirect product with a simply connected , nilpotent Lie group and a derivation of the Lie algebra , which satisfies the condition that all eigenvalues have positive real parts.
One speaks of a pure Heintze group when all eigenvalues are positive real numbers .
A pure Heintze group is of the Carnot type if the eigenspace for the smallest eigenvalue of generates the Lie algebra .
If is an Abelian group , the Heintze groups are called of the Abelian type .
properties
Heintze groups are solvable Lie groups. They have negative section curvature .
Examples
All symmetric spaces of non-compact type of rank 1 are Heintze groups, in particular the real-hyperbolic and the complex-hyperbolic space . These symmetrical spaces are obtained as the semi-direct product of a horosphere (which is a nilpotent Lie group) with .
Heintze's theorem
All homogeneous spaces with negative sectional curvature are Heintze groups.
Classification down to quasi-isometry
An open guess is that quasi-isometric Heintze groups are isomorphic. This conjecture has been proven for Heintze groups of the Carnot type and of the Abelian type.
literature
- Ernst Heintze : On homogeneous manifolds of negative curvature. Math. Ann. 211, 23-34, 1974.
- Ursula Hamenstädt : On the theory of Carnot-Carathéodory metrics and their applications. Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, 1986.
- Pierre Pansu : Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. of Math. (2), 129 (1), 1-60, 1989.
- Pierre Pansu: Dimension conforme et sphere à l'infini des variétés à courbure négative. Ann. Acad. Sci. Fenn. Ser. AI Math., 14 (2), 177-212, 1989.
- Xiangdong Xie : Large scale geometry of negatively curved . Geom. Topol., 18 (2), 831-872, 2014.
- Matias Carrasco, Emiliano Sequiera: On quasi-isometry invariants associated with the derivation of a Heintze group. , ArXiv