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

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 be a real Lie algebra. It is called a complete Lie algebra if each map

in its adjoint representation is hyperbolic, i.e., it has only real eigenvalues. Let G be a solvable Lie group whose Lie algebra is complete. Then for any closed subgroup of G, the solvmanifold 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, MR 0486307
  • 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], "Solvmanifold", Encyclopedia of Mathematics, EMS Press
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