User:Maximilian Janisch/latexlist/Algebraic Groups/Solv manifold

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solvmanifold, solvable manifold

A homogeneous space $N$ of a connected solvable Lie group $k$ (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 $N$.

Examples: $R ^ { n }$, the torus $T ^ { n }$, the Iwasawa manifold $N / I$ (where $M$ is the group of all upper-triangular matrices with 1's on the main diagonal in $GL ( 3 , R )$ and $1$ is the subgroup of all integer points in $M$), $K ^ { 2 }$ (the Klein bottle), and $M b$ (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 $R ^ { n }$, $T ^ { n }$, $N / I$, but not $K ^ { 2 }$ and $M b$). The following results are due to A.I. Mal'tsev (see [5]). 1) Every nil manifold $M = G / H$ is diffeomorphic to $M ^ { * } \times R ^ { n }$, where $M ^ { * }$ is a compact nil manifold. 2) If $N$ is compact and $k$ acts effectively on $N$, then the stabilizer $H$ is a discrete subgroup. 3) A nilpotent Lie group $k$ (cf. Lie group, nilpotent) acts transitively and locally effectively on some compact manifold if and only if its Lie algebra $8$ has a $0$-form. If, in addition, $k$ is simply connected, then it is isomorphic to a unipotent algebraic group defined over $0$ and $H$ is an arithmetic subgroup of $k$. 4) The fundamental group $\pi _ { 1 } ( M )$ of a compact nil manifold $N$ (which is isomorphic to $H$ when $k$ is simply connected and its action on $N$ 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 $N$ there is a solvmanifold $M$ which is a finitely-sheeted covering of it and is diffeomorphic to $M ^ { * } \times R ^ { n }$, where $M ^ { * }$ is some compact solvmanifold. An arbitrary solvmanifold cannot always be decomposed into a direct product $M ^ { * } \times R ^ { n }$, but it is diffeomorphic (see [1], [4]) to the space of a vector bundle over some compact solvmanifold (for $M b$ the corresponding bundle is a non-trivial line bundle over $s ^ { 1 }$). The fundamental group $\pi _ { 1 } ( M )$ of an arbitrary solvmanifold $N$ is polycyclic (cf. Polycyclic group), and if $N$ is compact, it determines $N$ uniquely up to a diffeomorphism. A group $31$ is isomorphic to $\pi _ { 1 } ( M )$ for some compact solvmanifold $N$ if and only if it is contained in an exact sequence of the form

\begin{equation} \{ e \} \rightarrow \Delta \rightarrow \pi \rightarrow Z ^ { s } \rightarrow \{ e \} \end{equation}

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 $N$. If a solvable Lie group $k$ acts transitively and locally effectively on a compact solvmanifold $M = G / H$, then $N$ is fibred over a torus with fibre $N / ( H \cap N )$, where $M$ is the nil radical of $k$. A solvmanifold $M = G / H$ is compact if and only if there is a $k$-invariant measure on $N$ with respect to which the volume of $N$ is finite.

Every solvmanifold $N$ 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)