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

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

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

References

• Auslander, Louis (1973), "An exposition of the structure of solvmanifolds. Part I: Algebraic theory" (PDF), Bulletin of the American Mathematical Society, 79 (2): 227–261, doi:10.1090/S0002-9904-1973-13134-9, MR 0486307
  • — (1973), "Part II: $G$-induced flows", Bull. Amer. Math. Soc., 79 (2): 262–285, doi:10.1090/S0002-9904-1973-13139-8, MR 0486308
• Cooper, Daryl; Scharlemann, Martin (1999), "The structure of a solvmanifold's Heegaard splittings" (PDF), Proceedings of 6th Gökova Geometry-Topology Conference, Turkish Journal of Mathematics, 23 (1): 1–18, ISSN 1300-0098, MR 1701636
• Gorbatsevich, V. V. (2001) [1994], "Solv manifold", Encyclopedia of Mathematics, EMS Press