Solv manifold

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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)
How to Cite This Entry:
Solv manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Solv_manifold&oldid=52493
This article was adapted from an original article by V.V. Gorbatsevich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article