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. ${\displaystyle N\rtimes _{\alpha }\mathbb {R} }$ ${\displaystyle N}$ ${\displaystyle \alpha \colon {\mathfrak {n}}\to {\mathfrak {n}}}$ ${\displaystyle {\mathfrak {n}}=Lie(N)}$ ${\displaystyle \alpha }$

One speaks of a pure Heintze group when all eigenvalues ​​are positive real numbers . ${\displaystyle \alpha }$

A pure Heintze group is of the Carnot type if the eigenspace for the smallest eigenvalue of generates the Lie algebra . ${\displaystyle \alpha }$ ${\displaystyle {\mathfrak {n}}}$

If is an Abelian group , the Heintze groups are called of the Abelian type . ${\displaystyle N}$ ${\displaystyle N\rtimes _{\alpha }\mathbb {R} }$

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 . ${\displaystyle \mathbb {R} }$

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 . ${\displaystyle \mathbb {R} ^{n}\rtimes _{\alpha }\mathbb {R} }$Geom. Topol., 18 (2), 831-872, 2014.
• Matias Carrasco, Emiliano Sequiera: On quasi-isometry invariants associated with the derivation of a Heintze group. , ArXiv