Flat Pseudo-Riemannian Homogeneous Spaces with Non-Abelian Holonomy Group

We construct homogeneous flat pseudo-Riemannian manifolds with non-abelian fundamental group. In the compact case, all homogeneous flat pseudo-Riemannian manifolds are complete and have abelian linear holonomy group. To the contrary, we show that there do exist non-compact and non-complete examples, where the linear holonomy is non-abelian, starting in dimensions $\geq 8$, which is the lowest possible dimension. We also construct a complete flat pseudo-Riemannian homogeneous manifold of dimension 14 with non-abelian linear holonomy. Furthermore, we derive a criterion for the properness of the action of an affine transformation group with transitive centralizer.


Introduction
A flat pseudo-Riemannian manifold M is called homogeneous if its isometry group acts transitively. As examples show [2,4], non-compact flat pseudo-Riemannian homogeneous manifolds manifolds are not necessarily complete. The study of complete flat homogeneous pseudo-Riemannian manifolds was pioneered by Wolf in a series of papers [8,9,10]. Such manifolds are isometric to a manifold of the form Ê r,s / Γ, for some subgroup Γ ≤ Iso(Ê r,s ). Homogeneity implies that the centralizer of Γ in Iso(Ê r,s ) acts transitively on Ê r,s . One basic observation is that in this situation the group Γ is nilpotent of nilpotency class at most two. This fact also holds for the holonomy group Γ of a non-complete homogeneous pseudo-Riemannian manifold. Apparently, it was believed for some time that Γ, or, which is considerably weaker, the linear part of Γ should be abelian. However, as observed in [1] non-abelian fundamental groups Γ appear for compact complete flat homogeneous pseudo-Riemannian manifolds.
In this note, we present some additional new results on the structure of flat pseudo-Riemannian homogeneous manifolds. Although non-abelian fundamental groups Γ do appear, in the compact case the linear holonomy is always abelian. In addition, we show that every homogeneous flat pseudo-Riemannian manifold of dimension less than eight has abelian linear holonomy.
As one our main results, we give examples of homogeneous manifolds with nonabelian linear holonomy group. We construct an eight-dimensional non-complete manifold U/ Γ 1 , where U is an open domain in Ê 4,4 , and a fourteen-dimensional complete manifold Ê 7,7 / Γ 2 , both with non-abelian linear holonomy. The groups Γ 1 ≤ Iso(Ê 4,4 ) and Γ 2 ≤ Iso(Ê 7,7 ) are isomorphic to the integral Heisenberg group on two generators and map injectively to their linear parts. These manifolds give the first examples of flat pseudo-Riemannian homogeneous manifolds with non-abelian linear holonomy group.

Preliminaries
Here, Ê r,s denotes Ê r+s endowed with a scalar product ·, · of signature (r, s), and Iso(Ê r,s ) its group of isometries. Affine maps of Ê r+s are written as γ = (I +A, v), where I +A is the linear part (I the identity matrix), and v the translation part.
The groups Γ ⊂ Iso(Ê r,s ) with transitive centralizer in Iso(Ê r,s ) were studied first in [8]. We sum up some of the results for later reference. Note that all of the following holds also if the centralizer of Γ is only required to have an open orbit in   For γ = (I + A, v) ∈ Γ, set hol(γ) = I + A (the linear component of γ). We write A = log(hol(γ)).
In the latest edition of the book [7], a characterization of those Γ with abelian linear holonomy is given: Proposition 2.6. The following are equivalent: (1) hol(Γ) is abelian.
The proof of Proposition 2.6 uses only the lemmata above. Thus, the following structure theorem for groups Γ, such that the centralizer of Γ has an open orbit in Ê r,s , and Γ with abelian linear holonomy, holds: Theorem 2.7. If hol(Γ) is abelian, then for every Witt basis with respect to U Γ (see section 4) and (I + A, v) ∈ Γ, the matrix A is of the form In section 3, we show that for compact M the holonomy group hol(Γ) is always abelian, and we present a more refined classification and structure theorem for the groups Γ in the compact case. We give a modification for the structure theorem, Theorem 2.7, which holds for arbitrary Γ, in section 4. In section 5, we show that groups Γ such that hol(Γ) is not abelian exist only for dimensions ≥ 8, and in section 6 we give an example of such a group. This group does not act freely and therefore cannot be the fundamental group of a complete flat homogeneous pseudo-Riemannian manifold M . But it gives rise to a non-complete example M . We also present an example of a group Γ which acts freely on Ê 7,7 and has a transitive centralizer. This group gives rise to a complete 14-dimensional homogeneous flat pseudo-Riemannian manifold with non-abelian linear holonomy group. To show that the groups Γ involved act properly we derive in section 7 a criterion which shows that a discrete unipotent group acting freely on Ê n , and whose centralizer has an open orbit, acts properly on Ê n .

Compact Flat Pseudo-Riemannian Spaces
In this section, let M be a compact flat homogeneous pseudo-Riemannian manifold. By [5] (see [1,Corollary 4.5] for an alternative proof), M must be complete.
Therefore, M = Ê r,s / Γ, for some group Γ ≤ Iso(Ê r,s ) which acts properly discontinuously and freely on Ê r,s . Let G be the centralizer of Γ in Iso(Ê r,s ). Since M is homogeneous and compact, then, as follows from [1,Theorem 4.6], G is a nilpotent Lie group which acts simply transitively on Ê r,s by isometries. Let x 0 ∈ Ê r,s be a fixed basepoint. There is a unique left invariant pseudo-Riemannian metric ·, · G on G such that the orbit map o : G → Ê r,s , g → g ·x 0 , is an isometry. Moreover, the metric ·, · G is biinvariant, see [1,Theorem 4.6]. The map o induces an isometry G/Γ → M , whereΓ is a lattice subgroup of G, which is isomorphic to Γ, and G/Γ inherits the pseudo-Riemannian structure from (G, ·, · G ). It also follows that G is (at most) two-step nilpotent (see [1,Lemma 4.8]). Such manifolds G/Γ necessarily have abelian linear holonomy group: Theorem 3.1. Let G be a Lie group with a biinvariant flat pseudo-Riemannian metric ·, · G , andΓ ≤ G be a lattice. Then the compact flat pseudo-Riemannian homogeneous manifold G/Γ has abelian linear holonomy.
Proof. Let ρ : G → Iso(Ê r,s ) be the development representation of the rightmultiplication of G and put Γ = ρ(Γ). Then, as above, there is an orbit map o : G → Ê r,s , which is an isometry and satisfies o(gγ) = ρ(γ)o(g). Let g denote the Lie algebra of G. By [1, Proposition 3.3, Lemma 5.10], the differential of ρ at the identity is equivalent to the affine representation X → ( 1 2 ad(X), X) of g on the vector space of g. In particular, the linear part of the differential of ρ is equivalent to the adjoint representation ad of g. Since g is twostep nilpotent, the adjoint representation ad has abelian image. It follows that the linear part of ρ(G) is abelian. Since Γ ≤ ρ(G), this implies that Γ has abelian linear part.
Remark 3.2. Let ·, · g denote the inner product induced on g by ·, · G . Biinvariance of ·, · G is equivalent to equals the commutator subalgebra of g. (The last equality holds because the elements in log(Γ) generate g, since Γ is a lattice in G.) Using biinvariance and 2-step nilpotency, it is easy to see that the space U Γ = [g, g] is totally isotropic. By Theorem 2.7, this is equivalent to hol(Γ) being abelian. To specify a biinvariant pseudo-Riemannian metric on the Lie group G it is equivalent to construct a biinvariant inner product ·, · g on the Lie algebra of g. The metric is flat if and only if g is two-step nilpotent. Below, we state a structure theorem for such pairs (g, ·, · g ), taken from [1, Theorem 5.15]. By the above, this yields a structure theorem for groups Γ which are the fundamental groups of compact homogeneous flat pseudo-Riemanninan manifolds.
Recall that for an abelian Lie algebra a and its dual a * , a Lie product on the space a ⊕ a * is given by where ad * denotes the coadjoint representation, and ω ∈ Z 2 (a, a * ) is a 2-cocycle for the adjoint representation. We use the notation g = a ⊕ ω a * for this Lie algebra. An inner product of split signature on g is defined by and it can be shown to be biinvariant if and only if the 3-form F ω (X 1 , X 2 , X 3 ) = ω(X 1 , X 2 ), X 3 on a is alternating and satisfies for all X i ∈ a. Then a ⊕ ω a * is a 2-step nilpotent Lie algebra with biinvariant inner product ·, · g . Theorem 3.4. Let g be a 2-step nilpotent Lie algebra with biinvariant inner product ·, · g . Then there exists an abelian Lie algebra a, an alternating 3-form F ω on a and an abelian Lie algebra z 0 such that g can be written as a direct product of metric Lie algebras is an isotropic subspace of g. Biinvariance shows that its orthogonal complement [g, g] ⊥ is the center z(g). Let a denote the isotropic subspace dual to [g, g] in g (then [g, g] can be identified with the dual space a * of a). Finally, let z 0 be a complement of a * in z(g), that is z(g) = a * ⊕ z 0 . Then z 0 commutes with and is orthogonal to a and a * . So g = (a ⊕ ω a * ) ⊕ z 0 for some 2-cocycle ω ∈ Z 2 (a, a * ).

Structure Theorem
Let Γ ⊂ Iso(Ê r,s ) such that its centralizer in Iso(Ê r,s ) has an open orbit in Ê r,s .
For short, we write ∆ for the center of Γ. This group is abelian, so it satisfies the conditions of Theorem 2.7. We set U Γ = γ∈Γ im A, U ∆ = γ∈∆ im A and U 0 = U Γ ∩ U ⊥ Γ , which is a totally isotropic subspace. It follows from Lemma 2.3 that It is easy to see that Proof. Let y ∈ U ⊥ ∆ . For all x ∈ Ê r,s and A, B ∈ log(hol(Γ)), Bx, Ay = − ABx, y = 0, by Lemma 2.3 and because AB is central. Hence Ay ⊥ U Γ , that is, Ay ∈ U 0 .
The following proposition sums up the above: is stabilized by log(hol(Γ)) such that each subspace is mapped to the next in the chain.
Given the totally isotropic subspace U 0 , we can find a Witt basis for Ê r,s with respect to U 0 as follows: If k = dim U 0 , there exists a basis for Ê r,s , {u 1 , . . . , u k , w 1 , . . . , w n−2k , u * 1 , . . . , u * k }, such that {u 1 , . . . , u k } is a basis of U 0 , {w 1 , . . . , w n−2k } is a basis of a non-degenerate subspace W such that U ⊥ 0 = U 0 ⊕ W , and {u * 1 , . . . , u * k } is a basis of a space U * 0 such that u i , u * j = δ ij (then U * 0 is called a dual space for U 0 ). LetĨ denote the signature matrix representing the restriction of ·, · to W with respect to the chosen basis of W .
The following generalizes Theorem 2.7: Theorem 4.4. Let γ = (I + A, v) ∈ Γ and fix a Witt basis with respect to U 0 . Then the matrix representation of A in this basis is with B ∈ Ê (n−2k)×k and C ∈ so k (where k = dim U 0 ). The columns of B are isotropic and mutually orthogonal with respect toĨ.
Proof. With respect to the given Witt basis, A is represented by a matrix The condition A 2 = 0 implies −B ⊤Ĩ B = 0, so all columns of B are isotropic and mutually orthogonal with respect toĨ.

Dimension Bounds for Non-Abelian Holonomy Groups
We sum up two rules which have to be satisfied by the representation matrices (4.3). Given matrices A i (i = 1, 2), B i and C i refer to the respective matrix blocks in (4.3).
(1) Crossover rule: Given A 1 and A 2 , let b i 2 be a column of B 2 and b k 1 a column of B 1 .
1Ĩ B 2 as the skew-symmetric upper right block, so its entries are the values − b k 1 , b i 2 .) (2) Duality rule: Assume A 1 is not central (that is A 1 A 2 = 0 for some A 2 ).
Then B 2 contains a column b i 2 and B 1 a column b j 1 such that b j 1 , b i 2 = 0.
As Example 6.2 shows, this is a sharp lower bound.
Proof. If hol(Γ) is not abelian, there exist Let W be a vector space complement of U 0 in U ⊥ 0 , so W is non-degenerate and x ∈ Ê r,s can be written x = u + w + u * with u ∈ U 0 , w ∈ W, u * ∈ U * 0 . Then By the duality rule, there are columns in B 1 , B 2 which are non-orthogonal to one another. Then, by the crossover rule, B 1 and B 2 together contain at least four linearly independent columns. This implies dim W ≥ 4.
, this means the skew-symmetric matrix C 3 is non-zero. Hence C 3 must have at least two columns, that is dim U 0 ≥ 2. Then Remark 5.2. With the additional assumption that the centralizer of Γ in Iso(Ê r,s ) acts transitively, the second author has a proof (to appear in his dissertation) that the dimension bound in Theorem 5.1 can be improved to n ≥ 14. As Example 6.4 shows, this is a sharp lower bound. Proof. By taking the Zariski closure, we may assume from the beginning that Γ is an algebraic subgroup of Iso(Ê r,s ) . Since the elements of Γ are unipotent, the algebraic group Γ is also connected. The centralizer G of Γ is also an algebraic subgroup, and as such it has finitely many open orbits in Ê r,s (cf. [1, Proposition
In the chosen basis, the pseudo-scalar product is represented by the matrix Q = where x = (x 1 , x 2 ) ⊤ , y = (y 1 , y 2 , y 3 , y 4 ) ⊤ , z = (z 1 , z 2 ) ⊤ are arbitrary and Hence the elements exp(S) are contained in the centralizer of Γ 4,4 in Iso(Ê r,s ). As x, y, z are arbitrary, the centralizer of Γ 4,4 has an open orbit U through the point 0. The set of all elements S is not a Lie subalgebra of the centralizer. Example 6.4. Let Γ 7,7 ⊂ Iso(Ê 7,7 ) be the group generated by One checks that A 2 i = 0 and that Γ 7,7 is isomorphic to a discrete Heisenberg group.
In the chosen basis, the pseudo-scalar product is represented by the matrix . The following elements S ∈ iso(Ê 7,7 ) commute with (A 1 , v 1 ) and (A 2 , v 2 ): where x = (x 1 , . . . , x 5 ) ⊤ , y = (y 1 , . . . , y 4 ) ⊤ , z = (z 1 , . . . , z 5 ) ⊤ are arbitrary and The linear part of such a matrix S is conjugate to a strictly upper triangular matrix via conjugation with the matrix T = (e 1 , e 2 , e 3 , e 4 , e 7 + e 8 , e 5 , e 6 , e 9 , e 10 , e 11 , e 12 , e 13 , e 14 , e 7 − e 8 ), where e i denotes the ith unit vector. Hence, the elements exp(S) generate a unipotent group of isometries whose translation parts contain all of Ê 14 . Therefore, the centralizer of Γ 7,7 in Iso(Ê 7,7 ) acts transitively (see [ It can be verified that the set of all matrices S forms a 3-step nilpotent Lie subalgebra of the centralizer algebra. Hence the set of all exp(S) forms a unipotent group of isometries acting simply transitively on Ê 7,7 . Corollary 6.5. There exists a flat complete homogeneous pseudo-Riemannian manifold of signature (7, 7) with non-abelian linear holonomy group.

Properness of Actions with Transitive Centralizer
Recall that an action of a Lie group L on a locally compact Hausdorff space X is called proper if and only if for all compact sets K ⊂ X the set {ℓ ∈ L | ℓK ∩ K = ∅} is compact. Proof. Choose a basepoint x 0 ∈ X such that H = G x0 is the stabilizer of x 0 . Then X is homeomorphic to G/H via the orbit map o : G/H → X, g → g · x 0 . The right-action of N G (H) on G induces a continuous homomorphism onto the centralizer Z X (G) of G in Diff(X). Let L denote the preimage of L in N G (H). In particular, if L is closed in Diff(X) then L is closed in G. Note that X/L = G/L is a Hausdorff space if and only if the subgroup L is closed in G. Since L acts freely on X, X/L is Hausdorff if and only if L acts properly on X. This proves the lemma.
We can apply this criterion in the affine situation, as follows: Proposition 7.2. Let L ≤ Aff(Ê n ) be a subgroup whose centralizer in Aff(Ê n ) acts transitively on Ê n . Then the action of L on Ê n is proper if and only if L is a closed subgroup of Aff(Ê n ).
Similarly, assume that the centralizer G of L in Aff(Ê n ) has an open orbit U = G · x 0 which is preserved by L. Then L acts freely on U , and the action is proper if and only if L is closed in Diff(U ). Since Diff(U ) ∩ Aff(Ê n ) is closed in Aff(Ê n ) (cf. [1, Lemma 6.9]), the above proposition generalizes to: Remark 7.4. Püttmann [6,Section 4.2] gives an example of a free action of the abelian group ( 2 , +) on 5 by unipotent affine transformations, such that the quotient is not a Hausdorff space. Hence the action is not proper.