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 [math]\displaystyle{ n=2 }[/math], 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 [math]\displaystyle{ \mathfrak{g} }[/math] be a real Lie algebra. It is called a complete Lie algebra if each map

[math]\displaystyle{ \operatorname{ad}(X)\colon \mathfrak{g} \to \mathfrak{g}, X \in \mathfrak{g} }[/math]

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

References

• Auslander, Louis (1973), "An exposition of the structure of solvmanifolds. Part I: Algebraic theory", Bulletin of the American Mathematical Society 79 (2): 227–261, doi:10.1090/S0002-9904-1973-13134-9, http://www.ams.org/bull/1973-79-02/S0002-9904-1973-13134-9 
  • Auslander, Louis (1973), "Part II: $G$-induced flows", Bull. Amer. Math. Soc. 79 (2): 262–285, doi:10.1090/S0002-9904-1973-13139-8, http://www.ams.org/bull/1973-79-02/S0002-9904-1973-13139-8 
• Cooper, Daryl; Scharlemann, Martin (1999), "The structure of a solvmanifold's Heegaard splittings", Turkish Journal of Mathematics 23 (1): 1–18, ISSN 1300-0098, http://journals.tubitak.gov.tr/math/issues/mat-99-23-1/mat-23-1-1-98071.pdf 
• Hazewinkel, Michiel, ed. (2001), "Solv manifold", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page