# 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 Malcev, 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 G. Mostow and proved by L. Auslander and R. 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.

## Odd section

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

- ad

in its adjoint representation is hyperbolic, i.e. has real eigenvalues. Let *G* be a solvable Lie group whose Lie algebra is complete. Then for any closed subgroup *Γ* of *G*, the solvmanifold *G*/*Γ* is a **complete solvmanifold**.

## References

- L. Auslander,
*An exposition of the structure of solvmanifolds*I, II, Bull. Amer. Math. Soc., 79:2 (1973), pp. 227–261, 262–285 - Cooper, Daryl; Scharlemann, Martin (1999), "Proceedings of 6th Gökova Geometry-Topology Conference",
*Turkish Journal of Mathematics*,**23**(1): 1–18, ISSN 1300-0098, MR 1701636`|chapter=`

ignored (help) - V.V. Gorbatsevich (2001), "S/s086100", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4