Solvmanifold

Examples

• A solvable Lie group is trivially a solvmanifold.
• Every nilpotent group is solvable, therefore, every nilmanifold is a solvmanifold. This class of examples includes n-dimensional tori and the quotient of the 3-dimensional real Heisenberg group by its integral Heisenberg subgroup.
• The Möbius band and the Klein bottle are solvmanifolds that are not nilmanifolds.
• The mapping torus of an Anosov diffeomorphism of the n-torus is a solvmanifold. For n=2, these manifolds belong to Sol, one of the eight Thurston geometries.

Properties

• A solvmanifold is diffeomorphic to the total space of a vector bundle over some compact solvmanifold. This statement was conjectured by George Mostow and proved by Louis Auslander and Richard Tolimieri.
• The fundamental group of an arbitrary solvmanifold is polycyclic.
• A compact solvmanifold is determined up to diffeomorphism by its fundamental group.
• Fundamental groups of compact solvmanifolds may be characterized as group extensions of free abelian groups of finite rank by finitely generated torsion-free nilpotent groups.
• Every solvmanifold is aspherical. Among all compact homogeneous spaces, solvmanifolds may be characterized by the properties of being aspherical and having a solvable fundamental group.

Completeness

Let \mathfrak{g} be a real Lie algebra. It is called a complete Lie algebra if each map

:\operatorname{ad}(X)\colon \mathfrak{g} \to \mathfrak{g}, X \in \mathfrak{g}

in its adjoint representation is hyperbolic, i.e., it has only real eigenvalues. Let G be a solvable Lie group whose Lie algebra \mathfrak{g} is complete. Then for any closed subgroup \Gamma of G, the solvmanifold G/\Gamma is a complete solvmanifold.

References