Solv manifold

solvmanifold, solvable manifold

A homogeneous space $ M $ of a connected solvable Lie group $ G $ (cf. Lie group, solvable). It can be identified with the coset space $ G / H $, where $ H $ is the stabilizer subgroup of some point of the manifold $ M $.

Examples: $ \mathbf R ^{n} $, the torus $ T ^{n} $, the Iwasawa manifold $ N / I $ (where $ N $ is the group of all upper-triangular matrices with 1's on the main diagonal in $ \mathop{\rm GL}\nolimits ( 3 ,\ \mathbf R ) $ and $ I $ is the subgroup of all integer points in $ N $), $ K ^{2} $ (the Klein bottle), and $ \mathop{\rm Mb}\nolimits $ (the Möbius band).

The first solvmanifolds studied were those in the narrower class of nil manifolds (cf. Nil manifold), that is, homogeneous spaces of nilpotent Lie groups (such as $ \mathbf R ^{n} $, $ T ^{n} $, $ N / I $, but not $ K ^{2} $ and $ \mathop{\rm Mb}\nolimits $). The following results are due to A.I. Mal'tsev (see [5]). 1) Every nil manifold $ M = G / H $ is diffeomorphic to $ M ^{*} \times \mathbf R ^{n} $, where $ M ^{*} $ is a compact nil manifold. 2) If $ M $ is compact and $ G $ acts effectively on $ M $, then the stabilizer $ H $ is a discrete subgroup. 3) A nilpotent Lie group $ G $ (cf. Lie group, nilpotent) acts transitively and locally effectively on some compact manifold if and only if its Lie algebra $ \mathfrak G $ has a $ \mathbf Q $-form. If, in addition, $ G $ is simply connected, then it is isomorphic to a unipotent algebraic group defined over $ \mathbf Q $ and $ H $ is an arithmetic subgroup of $ G $. 4) The fundamental group $ \pi _{1} (M) $ of a compact nil manifold $ M $ (which is isomorphic to $ H $ when $ G $ is simply connected and its action on $ M $ is locally effective) determines it up to a diffeomorphism. The groups $ \pi _{1} (M) $ that can arise here are just the finitely-generated nilpotent torsion-free groups.

These results admit partial generalizations to arbitrary solvmanifolds. Thus, for any solvmanifold $ M $ there is a solvmanifold $ M ^ \prime $ which is a finitely-sheeted covering of it and is diffeomorphic to $ M ^{*} \times \mathbf R ^{n} $, where $ M ^{*} $ is some compact solvmanifold. An arbitrary solvmanifold cannot always be decomposed into a direct product $ M ^{*} \times \mathbf R ^{n} $, but it is diffeomorphic (see [1], [4]) to the space of a vector bundle over some compact solvmanifold (for $ \mathop{\rm Mb}\nolimits $ the corresponding bundle is a non-trivial line bundle over $ S ^{1} $). The fundamental group $ \pi _{1} (M) $ of an arbitrary solvmanifold $ M $ is polycyclic (cf. Polycyclic group), and if $ M $ is compact, it determines $ M $ uniquely up to a diffeomorphism. A group $ \pi $ is isomorphic to $ \pi _{1} (M) $ for some compact solvmanifold $ M $ if and only if it is contained in an exact sequence of the form $$ \{ e \} \rightarrow \Delta \rightarrow \pi \rightarrow \mathbf Z ^{s} \rightarrow \{ e \} , $$ where $ \Delta $ is a finitely-generated nilpotent torsion-free group. Every polycyclic group has a subgroup of finite index that is isomorphic to $ \pi _{1} (M) $ for some compact solvmanifold $ M $. If a solvable Lie group $ G $ acts transitively and locally effectively on a compact solvmanifold $ M = G / H $, then $ M $ is fibred over a torus with fibre $ N / (H \cap N ) $, where $ N $ is the nil radical of $ G $. A solvmanifold $ M = G / H $ is compact if and only if there is a $ G $-invariant measure on $ M $ with respect to which the volume of $ M $ is finite.

Every solvmanifold $ M $ is aspherical (that is, the homotopy group $ \pi _{i} (M) = 0 $ for $ i \geq 2 $). Among all compact homogeneous spaces, compact solvmanifolds are characterized by asphericity and the solvability of $ \pi _{1} (M) $ (see [3]).

References

[1]	L. Auslander, "An exposition of the structure of solvmanifolds I, II" Bull. Amer. Math. Soc. , 79 : 2 (1973) pp. 227–261; 262–285 MR486308
[2]	L. Auslander, R. Szczarba, "Vector bundles over tori and noncompact solvmanifolds" Amer. J. Math. , 97 : 1 (1975) pp. 260–281 MR0383443 Zbl 0303.22006
[3]	V.V. Gorbatsevich, "On Lie groups, transitive on Solv manifolds" Math. USSR.-Izv. , 11 (1977) pp. 271–291 Izv. Akad. Nauk. SSSR Ser. Mat. , 41 (1977) pp. 285–307
[4]	G. Mostow, "Some applications of representative functions to solvmanifolds" Amer. J. Math. , 93 : 1 (1971) pp. 11–32 MR0283819 Zbl 0228.22015
[5]	M. Raghunatan, "Discrete subgroups of Lie groups" , Springer (1972)