Solvmanifold

In mathematics, a solvmanifold is a homogeneous space of a connected solvable Lie group. It may also be characterized as a quotient of a connected solvable Lie group by a closed subgroup. (Some authors also require that the Lie group be simply-connected, or that the quotient be compact.) A special class of solvmanifolds, nilmanifolds, was introduced by Anatoly Maltsev, who proved the first structural theorems. Properties of general solvmanifolds are similar, but somewhat more complicated.

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.