Discrete Cocompact Subgroups of the Five-Dimensional Connected and Simply Connected Nilpotent Lie Groups

The discrete cocompact subgroups of the five-dimensional connected, simply connected nilpotent Lie groups are determined up to isomorphism. Moreover, we prove if $G=N\times A$ is a connected, simply connected, nilpotent Lie group with an Abelian factor $A$, then every uniform subgroup of $G$ is the direct product of a uniform subgroup of $N$ and ${\mathbb Z}^r$ where $r=\dim A$.


Introduction
To solve many problems in various branches of mathematics, for example, in differential geometry (invariant geometric structures on Lie groups), in spectral geometry, in physics (multidimensional models of space-times) etc. (see [4]), we are led to study the explicit description of uniform subgroups (i.e., discrete cocompact) of solvable Lie groups. Towards that purpose, we focus attention to the description of uniform subgroups of a connected, simply connected non Abelian nilpotent Lie groups satisfying the rationality criterion of Malcev (see [9]). At present we have no progress on this difficult problem although the classification for certain groups is already exists. To attack this problem, we will be in a first step interested in giving an explicit description of uniform subgroups of nilpotent Lie groups of dimension less or equal to 6. Another motivation comes from [13, p. 339]. For the case of 3 or 4 dimensions, the classification is given in [1,15,11]. For 5-dimensional nilpotent Lie groups, there are eight real connected and simply connected non Abelian nilpotent Lie groups which are G 3 × R 2 , G 4 × R and G 5,i for 1 ≤ i ≤ 6 [3]. The uniform subgroups of G 3 ×R 2 , G 5,1 , G 5,3 and G 5,5 are determined respectively in [15,5,12,6]. The aim of this paper is to complete the classification of the discrete cocompact subgroups of the Lie groups G 4 × R, G 5,2 , G 5,4 and G 5, 6 . This paper is organized as follows. In Section 2, we fix some notation which will be of use later and we record few standard facts about rational structures and uniform subgroups of a connected, simply connected nilpotent Lie groups. Section 3 is devoted to the study of the rationality of certain subalgebras of a given nilpotent Lie algebra with a rational structure. In Section 4, in order to determine the uniform subgroups of G 4 × R, we study a more general class when G is a connected, simply connected, nilpotent Lie group with an Abelian factor, that is G = N × A where A is an Abelian normal subgroup of G. Theorem 5 shows that every uniform subgroup Γ of G is a direct product of a uniform subgroup of N and Z r where r = dim(A). As an immediate application of this result, we determine the uniform subgroups of G 4 × R. We determine afterwards all uniform subgroups of the Lie groups G 5,2 , G 5,4 and G 5,6 .

Notations and basic facts
The aim of this section is to give a brief review of certain results from rational structures and uniform subgroups of connected and simply connected nilpotent Lie groups which will be needed later. The reader who is interested in detailed proof is referred to standard texts [2,16,9].

Rational structures and uniform subgroups
Let G be a nilpotent, connected and simply connected real Lie group and let g be its Lie algebra. We say that g (or G) has a rational structure if there is a Lie algebra g Q over Q such that g ∼ = g Q ⊗ R. It is clear that g has a rational structure if and only if g has an R-basis {X 1 , . . . , X n } with rational structure constants.
Let g have a fixed rational structure given by g Q and let h be an R-subspace of g. Define h Q = h ∩ g Q . We say that h is rational if h = R-span {h Q }, and that a connected, closed subgroup H of G is rational if its Lie algebra h is rational. The elements of g Q (or G Q = exp(g Q )) are called rational elements (or rational points) of g (or G).
A discrete subgroup Γ is called uniform in G if the quotient space G/Γ is compact. The homogeneous space G/Γ is called a compact nilmanifold. A proof of the next result can be found in Theorem 7 of [9] or in Theorem 2.12 of [16].
Theorem 1 (The Malcev rationality criterion). Let G be a simply connected nilpotent Lie group, and let g be its Lie algebra. Then G admits a uniform subgroup Γ if and only if g admits a basis {X 1 , . . . , X n } such that where the constants c ijα are all rational. (The c ijα are called the structure constants of g relative to the basis {X 1 , . . . , X n }.) More precisely, we have, if G has a uniform subgroup Γ, then g (hence G) has a rational structure such that g Q = Q-span {log(Γ)}. Conversely, if g has a rational structure given by some Q-algebra g Q ⊂ g, then G has a uniform subgroup Γ such that log(Γ) ⊂ g Q (see [2,9]). If we endow G with the rational structure induced by a uniform subgroup Γ and if H is a Lie subgroup of G, then H is rational if and only if H ∩ Γ is a uniform subgroup of H. Note that the notion of rational depends on Γ.

Weak and strong Malcev basis
Let g be a nilpotent Lie algebra and let B = {X 1 , . . . , X n } be a basis of g. We say that B is a weak (resp. strong) Malcev basis for g if g i = R-span {X 1 , . . . , X i } is a subalgebras (resp. an ideal) of g for each 1 ≤ i ≤ n (see [2]).
The lower central series (or the descending central series) of g is the decreasing sequence of characteristic ideals of g defined inductively as follows The characteristic ideal C 2 (g) = [g, g] is called the derived ideal of the Lie algebra g and denoted by D(g). Let D(G) = exp(D(g)), observe that we have D(G) = [G, G]. The Lie algebra g is called k-step nilpotent Lie algebra or nilpotent Lie algebra of class k if there is an integer k such that and We denote the center of G by Z(G) and the center of g by z(g).
Proposition 1 ( [10,2]). If g has rational structure, all the algebras in the descending central series are rational.
Let G be a group. The center Z(G) of G is a normal subgroup. Let C 2 (G) be the inverse image of Z(G/Z(G)) under the canonical projection G −→ G/Z(G). Then C 2 (G) is normal in G and contains Z(G). Continue this process by defining inductively: C 1 (G) = Z(G) and C i (G) is the inverse image of Z(G/C i−1 (G)) under the canonical projection G −→ G/C i−1 (G). Thus we obtain a sequence of normal subgroups of G, called the ascending central series of G.

Proposition 2 ([2]
). If G is a nilpotent Lie group with rational structure, all the algebras in the ascending central series are rational. In particular, the center z(g) of g is rational.
A proof of the next result can be found in Proposition 5.3.2 of [2]. A rational structure on g induces a rational structure on the dual space g * (for further details, see [2,Chaper 5]). If g has a rational structure given by the uniform subgroup Γ, a real linear functional f ∈ g * is rational (f ∈ g * Q , g Q = Q-span {log(Γ)}) if f, g Q ⊂ Q, or equivalently f, log(Γ) ⊂ Q. Let Aut(G) (respectively Aut(g)) denote the group of automorphism of G (respectively g). If ϕ ∈ Aut(G), ϕ * will denote the derivative of ϕ at identity. The mapping Aut(G) −→ Aut(g), ϕ −→ ϕ * is a groups isomorphism (since G is simply connected).
Theorem 2 ([9, Theorem 5]). Let G 1 and G 2 be connected simply connected nilpotent Lie groups and Γ 1 , Γ 2 uniform subgroups of G 1 and G 2 . Any abstract group isomorphism f between Γ 1 and Γ 2 extends uniquely to an isomorphism f of G 1 on G 2 ; that is, the following diagram We conclude this review with two results.

Smith normal form
A commutative ring R with identity 1 R = 0 and no zero divisors is called an integral domain. A principal ideal ring which is an integral domain is called a principal ideal domain.
Theorem 3 (elementary divisors theorem). If A is an n×n matrix of rank r > 0 over a principal ideal domain R, then A is equivalent to a matrix of the form where L r is an r×r diagonal matrix with nonzero diagonal entries d 1 , . . ., d r such that d 1 |d 2 |· · ·|d r .
The notation d 1 |d 2 | · · · |d r means d 1 divides d 2 , d 2 divides d 3 , etc. The elements d 1 , . . . , d r are called the elementary divisors of A. See [7] for a more precise result.

Hermite normal form
Definition 1 (Hermite normal form). A matrix (a ij ) ∈ Mat(m, n, R) with m ≤ n is in Hermite normal form if the following conditions are satisfied: (1) a ij = 0 for i > j; (2) a ii > 0 for i = 1, . . . , m; (3) 0 ≤ a ij < a ii for i < j.

Rationality of certain subalgebras
The following generalizes the Proposition 5.2.4 of [2]. Proof . This follows immediately from Lemma 1.14, Theorem 2.1 of [16] and Theorem 4.5 of [14].
Next, we prove the following proposition which will play an important role below.
Proof . Let {e 1 , . . . , e n } be a weak Malcev basis for g strongly based on Γ passing through h (see [2,Proposition 5.3.2]). We note h = R-span {e 1 , . . . , e p }. Let {e * 1 , . . . , e * n } be the dual basis of {e 1 , . . . , e n }. For i = 1, . . . , n and j = 1, . . . , p, we define The functionals f ij are rational. Then the kernels ker(f ij ) are rational subspaces in g (see [ 4 Uniform subgroups of nilpotent Lie group with an Abelian factor. Uniform subgroups of G 4 × R The aim of this section is to describe the classification of the uniform subgroups of G 4 × R. Let us begin with a more general situation. First, we introduce the following definition. Definition 2 ( [8,4]). An Abelian factor of a Lie algebra g is an Abelian ideal a for which there exists an ideal n of g such that g = n ⊕ a (i.e., [n, a] = {0}).
Let m(g) denote the maximum dimension over all Abelian factors of g. If z(g) is the center of g then the maximal Abelian factors are precisely the linear direct complements of z(g) ∩ D(g) in z(g), that is, those subspaces a ⊂ z(g) such that z(g) = z(g) ∩ D(g) ⊕a. Therefore Let g be a nilpotent Lie algebra and a an Abelian factor of g. Let g = n ⊕ R r be any decomposition in ideals of g. The next simple lemma establishes a relation between the derived algebra of g and the derived algebra of n. Its value will be immediately apparent in the proof of Theorem 5. Lemma 1. With the above notation, we have since a ⊂ z(g). Therefore D(g) = D(n). The proof of the lemma is complete.
In the sequel, the symbol ≃ denotes abstract group isomorphism.
Theorem 5. Let g be a nilpotent Lie algebra with maximal Abelian factor of dimension m(g) = r and let g = n ⊕ R r be any decomposition in ideals, that is R r is a maximal Abelian factor of g. Then we have the following: (1) The group G = exp(g) admits a uniform subgroup if and only if N = exp(n) admits a uniform subgroup.
(2) If Γ is a uniform subgroup of G, then there exists a uniform subgroup H of N such that Conversely, we suppose that G admits a uniform subgroup Γ. Let {e 1 , . . . , e n } be a strong Malcev basis for g strongly based on Γ passing through D(g) and D(g) +z(g). Put For every i = q + r + 1, . . . , n, we note where u i ∈ n and v i ∈ a. It is clear that {e 1 , . . . , e q , u q+r+1 , . . . , u n } is a basis for n and has the same structure constants as {e 1 , . . . , e n }. By the Malcev rationality criterion, N admits a uniform subgroup.
Since z(g) is Abelian, we can assume that (e 1 , . . . , e p ) are independent modulo a, i.e., they span a complement of a in z(g). Let for every i = p + 1, . . . , n where a i ∈ a and b i ∈ n. On the other hand, we have D(g) +z(g) = D(g) ⊕a, then a = R-span {a p+1 , . . . , a p+r }. Let Φ * : g −→ g be the function defined by We will show that Φ * is a Lie automorphism of g. In fact, if i ≤ p + r or j ≤ p + r, we have Φ * ([e i , e j ]) = [Φ * (e i ), Φ * (e j )] = 0. Next, we suppose that i, j > p + r. It is easy to verify that On the other hand, we have Consequently, we obtain Evidently, H = This completes the proof of the theorem.
A consequence of the above theorem, we deduce the uniform subgroups of G 4 ×R. We recall that the Lie algebra g 4 of G 4 is spanned by the vectors X 1 , . . . , X 4 such that the only non vanishing brackets are Corollary 1. Let {e} be the canonical basis of R. Every uniform subgroup Γ of G 4 × R has the following form where p 1 , p 2 , p 3 are integers satisfying p 1 > 0, p 2 > 0, p 1 p 2 + p 3 ∈ 2Z and 0 ≤ p 3 < 2p 1 . Furthermore, different choices for the p's give non isomorphic subgroups.
Proof . The proof follows from Theorem 5 and [6, Proposition 4.1] (see also [11,Theorem 1] 5 Uniform subgroups of G 5,2 Let g 5,2 be the two-step nilpotent Lie algebra with basis and non-trivial Lie brackets defined by Let G 5,2 be the corresponding connected and simply connected nilpotent Lie group. First, we introduce the following set: Theorem 6. 1. Let r = (r 1 , r 2 ) ∈ D 2 . Then is a uniform subgroup of G 5,2 .
3. For r and s in D 2 , Γ r and Γ s are isomorphic groups if and only if r = s.
Proof . The proof of this theorem will be achieved through a sequence of partial results.
Lemma 2. The group Aut(g 5,2 ) is the set of all matrices of the form Proof . It is easy to verify that R-span {X 1 , . . . , X 4 } is the unique one co-dimensional Abelian ideal of g. Therefore any automorphism A of g leaves invariant the subalgebras R-span {X 1 , X 2 } and R-span {X 1 , . . . , X 4 }. The remainder of the proof follows from (1).

Lemma 3.
Every uniform subgroup Γ of G 5,2 has the following form: there are integers p, q and α satisfying p, q > 0 and 0 ≤ α < q such that Proof . Let G = G 5,2 , g = g 5,2 and let Γ be a uniform subgroup of G. Let be the layer of the generic coadjoint orbits (see [2, Chapter 3]). As g * Q is dense in g * and Ω is a non-empty Zariski open set in g * then g * Q ∩ Ω = ∅. Let l ∈ g * Q ∩ Ω. Since l is rational then by Proposition 5.2.6 of [2], the radical g(l) of the skew-symmetric bilinear form B l on g defined by B l (X, Y ) = l, [X, Y ] (X, Y ∈ g), is also rational. It follows by Proposition 5 that c(g(l)) is also rational subalgebra of g. As then a simple calculation shows that c(g(l)) = R-span {X 1 , . . . , X 4 } .
On the other hand, the derived ideal [g, g] = R-span {X 1 , X 2 } is also rational subalgebra in g (see [ where a ij ∈ R. The mapping (Φ 1 ) * : g −→ g defined by is a Lie algebra automorphism. Then On the other hand, let the Lie algebra automorphism (Φ 2 ) * : g −→ g defined by It follows that there exist a, b, c, d ∈ R such that Next, since exp(X 5 ) exp(X 3 ) exp(−X 5 ) ∈ Γ 2 , then exp(X 1 ) ∈ Γ 2 and hence the ideal R-span{X 1 } is rational relative to Γ 2 . It follows that there exist x, y 1 , y 2 ∈ R such that Γ 2 = exp(Z(xX 1 )) exp(Z(y 1 X 1 + y 2 X 2 )) 5 i=3 exp(ZX i ).
Also, we use that exp(X 1 ) belongs to Γ 2 , we deduce that there exists p ∈ N * such that x = 1 p . Similarly, since exp(X 5 ) exp(X 4 ) exp(−X 5 ) ∈ Γ 2 , then exp(X 2 ) ∈ Γ 2 . Therefore there exists (q, m) ∈ N * × Z such that y 2 = 1 q and y 1 = − m pq . Then Finally, we observe that, if α is the remainder of the division of m by q, then
Finally, an appeal to Lemma 4 and Smith normal form completes the proof of Lemma 3.
6 Uniform subgroups of G 5,4 Let g 5,4 be the three-step nilpotent Lie algebra with basis B = (X 1 , . . . , X 5 ) with Lie brackets are given by and the non-defined brackets being equal to zero or obtained by antisymmetry. Let G 5,4 be the corresponding connected and simply connected nilpotent Lie group. For u ∈ Z, let and where m ∈ N * , the block matrix D = (α ij ; 1 ≤ i, j ≤ 3) is an upper-triangular integer invertible matrix and SL(3, Z) is the set of all integer matrices with determinant 1. We are ready to formulate our result. In the proof of the theorem we need the following lemma. Proof . Any automorphism A of g 5,4 leaves invariant the subalgebras R-span {X 1 , X 2 } and R-span {X 1 , X 2 , X 3 }. The remainder of the proof follows from (3).
Proof of Theorem 7. Let G = G 5,4 and g = g 5,4 . Assertion 1 is obvious. To prove the second, let Γ be a uniform subgroup of G. Since the ideals z(g) = R-span {X 1 , X 2 } and D(g) = R-span {X 1 , X 2 , X 3 } are rational, then there exists a strong Malcev basis B ′ of g strongly based on Γ and passing through z(g) and D(g). Let be the change of basis matrix from the basis B to the basis B ′ . Let Φ * ∈ Aut(g 5,4 ) defined by Φ * (X 5 ) = a 55 X 5 + a 45 X 4 + a 35 X 3 + a 25 X 2 + a 15 X 1 , Φ * (X 4 ) = a 54 X 5 + a 44 X 4 + a 34 X 3 + a 24 X 2 + a 14 X 1 .
The above equation may obviously rewritten as Using a similar technique, we obtain that there exist integer coefficients x 1 , x 2 , y 1 , y 2 satisfy x 2 1 + x 2 2 = 0 and y 2 1 + y 2 2 = 0 such that and b 33 X 1 = y 1 e 1 + y 2 e 2 .
We see that the conditions (9) This proves (4). The proof of the second equality (5) is shown similarly (repeat the proof of (4) verbatim, using mX 4 instead of X 5 ). The proof of theorem is complete.
7 Uniform subgroups of G 5,6 Let g 5,6 be the three-step nilpotent Lie algebra with basis with Lie brackets are given by and the non-defined brackets being equal to zero or obtained by antisymmetry. Let G 5,6 be the simply connected Lie group with Lie algebra g 5, 6 . Let We now state the final result of this paper. We first state and prove two lemmas.
Lemma 6. The following ideals are rational with respect to any rational structure on g 5,6 .
We conclude from Proposition 5 that a 4 is rational. The remainder of the proof follows from (13).
Proof of Theorem 8. Assertion 1. is obvious. To prove the second, let Γ be a uniform subgroup of G 5,6 . Since for every i = 1, . . . , 4, the ideal a i = R-span {X 1 , . . . , X i } is rational, then Finally, we achieve with the proof of 3. We show both directions. Let a ∈ Q * such that D, m D ′ , m ′ −1 = diag [a 5 , a 4 , a 3 , a 2 , a].
This completes the proof.
Remark 2. In the statement 2 of Theorem 8, the element D, m of B 6 is not unique.