Journal of Lie Theory Volume 11 (2001) 1–8 ◯^{C}2001 Heldermann Verlag

Maximal semigroups in the divisible hull of lattices in nilpotent Lie groups

Osvaldo Germano do Rocio

Communicated by J. D. Lawson

Abstract. We consider subsemigroups in the divisible hull of lattices in nilpotent Lie groups. Our principal result shows that such a semigroup is a group provided it is not contained in a proper semigroup with non-empty interior.

1. Introduction

A subsemigroup S of a group G is said to maximal if it is not a group and there is no semigroup T ≠ S containing S properly. When studying subsemigroups of groups many questions have a natural formulation and solution by means of the knowledge of maximal semigroups. This makes the problem of determining the maximal semigroups one of the major problems in the theory of semigroups.

Let N be a nilpotent Lie group. In this paper we look at the subsemigroups of the subgroups of N which are obtained by divisible extension of a lattice in the following sense. As it is well known N is a divisilbe group. If N is simply connected and Γ is a lattice in N we look at the semigroups in the smallest divisible group of N containing Γ. In this setting a simply connected N admits a lattice Γ if and only if there exists a basis, say β, of the Lie algebra 𝔫 of N, whose constants of structure are rational (see Raghunathan [5, Thm. 2.12]). Let 𝔫_{ℚ} denote the rational subspace spanned by β. If we identify N with 𝔫 through the exponential map, the group product in 𝔫 is provided by the Campbell-Hausdorff formula, which has constant coefficients. Hence 𝔫_{ℚ} becomes a subgroup of N which contains Γ. We show that 𝔫_{ℚ} is the smallest divisible subgroup of N that contains Γ.

The purpose of this paper is to characterize the maximal semigroups of 𝔫_{ℚ}. The main results show that S ⊂ 𝔫_{ℚ} is a maximal semigroup if and only if it is the intersection of a maximal semigroup with non-void interior in N with 𝔫_{ℚ}. Moreover, any semigroup in 𝔫_{ℚ} which is not a group is contained in a maximal semigroup. These results extend those of [6], [7], where the maximal semigroups

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag in the lattices of nilpotent Lie groups where given analogous characterizations. Actually, our basic technique comes from the results for semigroups in lattices together with the fact that a rational subgroup is a union of lattices.

2. The Divisible Hull of a Lattice in a Simply Connected Nilpotent Lie Group

In what follows we identify the simply connected nilpotent Lie group with its Lie algebra 𝔫 through the exponential map

exp : 𝔫 ⟶ N,

which is a diffeomorphism. Through this identification the group structure in N is carried back to 𝔫. We denote by ∗ the group product in 𝔫. It is given by the well known Campbell-Hausdorff formula, whose first terms are

X ∗ Y = X + Y + \frac{1}{2}[X, Y] + ⋯ X, Y ∈ 𝔫.

Since X^{n} = nX for all n ∈ ℤ and X ∈ 𝔫, it follows that N is a divisible and torsion free group. Hence if H is a subgroup of a nilpotent simply connected Lie group N, there exists a smallest divisible subgroup Div(H ) of N containing H.

Let Γ be a lattice in a simply connected nilpotent Lie group and let 𝔫_{ℚ}(Γ) be the ℚ-span of exp^{−1}(Γ). By theorem 2.12 in [5], 𝔫_{ℚ}(Γ) is a ℚ-Lie algebra of 𝔫. If we identify N with (𝔫, ∗) we have:

Lemma 2.1. 𝔫_{ℚ}(Γ) is a divisible subgroup of N.

Proof. If X ∈ 𝔫_{ℚ}(Γ) and n ∈ ℤ − {0} then X = r_{1} X_{1} + ⋯ + r_{k}X_{k} ,where X_{1}, ⋯ X_{k} ∈ Γ and r_{1}, ⋯, r_{k} ∈ ℚ. Let us suppose that r_{i} = a_{i}/b_{i}, where a_{i}, b_{i} ∈ ℤ for 1 ≤ i ≤ k and put s_{i} = a_{i}/nb_{i}. Then S = s_{1}X_{1} + ⋯ s_{k}X_{k} ∈ 𝔫_{ℚ}(Γ) and S^{n} = nS = X.

Let 𝔫^{l} the l-th term of the lower central series of 𝔫, that is, 𝔫^{0} = 𝔫 and 𝔫^{l} = [𝔫, 𝔫^{l−1}]. Let k such that 𝔫^{k} ≠ {0} and 𝔫^{k+1} = {0}.

Proposition 2.2. Let Γ be a lattice in a nilpotent simply connected Lie group N. Then 𝔫_{ℚ}(Γ) = Div(Γ).

Proof. Since 𝔫_{ℚ}(Γ) is a divisible subgroup we have that Div(Γ) ⊂ 𝔫_{ℚ}(Γ). In order to show the other inclusion we will argue by induction on dim N. If dim N = 1 then N is abelian and the result is trivial. If π : N ⟶ N/[N, N] is the natural map, then π(Γ) is a lattice in N/[N, N]. The group N/[N, N] being a real vector space, π(Γ) is the ℤ-span of a basis e_{1}, e_{2}, ⋯, e_{l} of this vector space. Let V be the ℝ -span of e_{1}, e_{2}, ⋯, e_{l−1} and put N′ = π^{−1}(V). Let 𝔫′ be the Lie algebra of N′. Then 𝔫′ may be identified with a subalgebra of 𝔫 of codimension 1. Now, by induction hypothesis, Div(Γ ∩ N′) = 𝔫′_{ℚ}(Γ ∩ N′). According to the proof of Theorem 2.12 in [5], 𝔫_{ℚ}(Γ) is a ℚ-Lie subalgebra, so there exists v ∈ exp^{−1}(Γ) such that

𝔫_{ℚ}(Γ) = ℚv + 𝔫′_{ℚ}(Γ ∩ N′)

and

[v, w] ∈ 𝔫′_{ℚ}(Γ ∩ N′) , ∀w ∈ 𝔫′_{ℚ}(Γ ∩ N′).

Then we have

[ℚv, 𝔫′_{ℚ}(Γ ∩ N′)] ⊂ 𝔫′_{ℚ}(Γ ∩ N′) = Div (Γ ∩ N′).

Let r ∈ ℚ and y ∈ 𝔫′_{ℚ}(Γ ∩ N′). Since 𝔫′_{ℚ}(Γ ∩ N′) = Div(Γ ∩ N′) ⊂ Div(Γ) and ℚv ⊂ Div(Γ) then rv ∗ y ∈ Div(Γ). By the Campbell-Hausdorff formula we get:

rv ∗ y = rv + y + z_{1},

where z_{1} ∈ Div(Γ ∩ N′) ∩ 𝔫^{1} .

Since (rv ∗ y) ∗ z_{1}^{−1} ∈ Div(Γ) we have that

(rv ∗ y) ∗ z_{1}^{−1} = (rv + y + z_{1}) ∗ z_{1}^{−1} = rv + y + z_{2},

where z_{2} ∈ Div(Γ ∩ N′) ∩ 𝔫^{2} .

Repeating the above argument j times we obtain z_{1}, z_{2}, ⋯ z_{j}, z_{j+1} such that

(rv ∗ y) ∗ z_{1}^{−1} ∗ z_{2}^{−1} ∗ ⋯ ∗ z_{j}^{−1} ∈ Div(Γ)

and

(rv ∗ y) ∗ z_{1}^{−1} ∗ z_{2}^{−1} ∗ ⋯ ∗ z_{i}^{−1} = rv + y + z_{i+1},

where z_{i} ∈ Div(Γ ∩ N′) ∩ 𝔫^{i} for 1 ≤ i ≤ j + 1. But 𝔫 is nilpotent, so 𝔫^{k+1} = {0} for some k ∈ ℕ. Thus

(rv ∗ y) ∗ z_{1}^{−1} ∗ z_{2}^{−1} ∗ ⋯ ∗ z_{k}^{−1} = rv + y ∈ Div (Γ).

This shows that 𝔫_{ℚ}(Γ) ⊂ Div(Γ). Therefore 𝔫_{ℚ}(Γ) = Div(Γ) .

Corollary 2.3. Let Γ_{1} and Γ be lattices of N such that Γ_{1} ⊂ Div(Γ). Then Div(Γ_{1} ) = Div(Γ).

Proof. Since Γ_{1} ⊂ Γ_{1} ⊂ Div(Γ), by Proposition 2.2 we have that

𝔫_{ℚ}(Γ_{1}) = Div (Γ_{1}) ⊂ Div (Γ) = 𝔫_{ℚ}(Γ).

By Theorem 2.12 of [5], 𝔫_{ℚ}(Γ_{1}) ⊗_{ℚ} ℝ is isomorphic to 𝔫_{ℚ}(Γ) ⊗_{ℚ} ℝ. Then 𝔫_{ℚ}(Γ_{1}) = 𝔫_{ℚ}(Γ) and Div(Γ_{1} ) = Div(Γ).

Lemma 2.4. If Γ is a lattice of N then Div(Γ) is a union of lattices of N.

Proof. Suppose that β = {α_{1}, α_{2}, ⋯ α_{n}} is a basis of 𝔫_{ℚ}(Γ). Then it is clear that

𝔫_{ℚ}(Γ) = ⋃_{(k_{1},... ,k_{n})∈ℕ^{n}} \frac{1}{k_{1}} ℤα_{1} + \frac{1}{k_{2}} ℤα_{2} + ⋯ + \frac{1}{k_{n}} ℤα_{n}

and for (k_{1}, ⋯, k_{n}) ∈ ℕ^{n},

Γ_{(k_{1},··· ,k_{n})} = \frac{1}{k_{1}} ℤα_{1} + \frac{1}{k_{2}} ℤα_{2} + ⋯ + \frac{1}{k_{n}} ℤα_{n}

is a lattice in the vector space 𝔫. But the proof of Theorem 2.12 in [5], shows that if Γ_{1} ⊂ 𝔫_{ℚ}(Γ) is a lattice in the vector space 𝔫 then the subgroup of N generates by Γ_{1} is a lattice of Ṅ. Then Γ_{(k_{1},··· ,k_{n})} generated a lattice ⟨Γ_{(k_{1},··· ,k_{n})}⟩ of N which is contained in 𝔫_{ℚ}(Γ).

We prove below that any finitely generated subgroup of a rational group is discrete. By a rational subspace of a vector space V we mean the space spanned over ℚ by a discrete subgroup in V .

Lemma 2.5. Let V be a finite dimensional vector space over ℝ. Then, every finitely generated subgroup of a ℚ-subspace W in V is a discrete subgroup of V .

Proof. We may assume that V = ℝ^{n} and W = ℚ^{n}. Let H be a subgroup of ℝ^{n} generated by X_{1}, X_{2}, …, X_{l} in ℚ^{n}, where l is a non negative integer. In this case there is an integer d such that dX_{i} ∈ ℤ^{n} for 1 ≤ i ≤ l. Hence dH is a subgroup of ℤ^{n}, showing that H is a discrete subgroup of ℝ^{n}.

Proposition 2.6. If Γ is a lattice of N then every finitely generated subgroup of Div(Γ) is a discrete subgroup.

Proof. By proposition 2.2 Div(Γ) = 𝔫_{ℚ}(Γ). Let β be a basis of 𝔫_{ℚ}(Γ) and take x_{1}, …, x_{l} ∈ R, where l is a non negative integer. By Lemma 2.5 the subgroup H of the vector space 𝔫 generated by x_{1}, …, x_{l} is discrete in 𝔫. By the proof of Theorem 2.12 in [5] this implies that H generates a discrete subgroup of N.

3. Semigroups in Div(Γ)

A standard assumption in the control theory of Lie groups is that semigroups under consideration have non-void interior. A controllability criterion is provided by the fact that a neighborhood of the identity in a topological group generates, as a semigroup, its identity component.

In order to avoid repetition we make the following technical definition:

Definition 3.1. A subset X of a topological group G is said to have property (C) if it is not contained in a proper subsemigroup of G with interior points.

This is a kind of controllability condition because it means that the only semigroup with non-void interior containing X is G itself.

For a subsemigroup S of a simply connected nilpotent Lie group N we let

S̃ = {t_{1}x_{1} ∗ t_{2}x_{2} ∗ ⋯ ∗ t_{l}x_{l} : t_{i} ≥ 0, x_{i} ∈ S, arbitrary l}

be the semigroup generated by the real positive powers of the elements in S. It is a ray semigroup in N and S is a set of infinitesimal generators of S̃.

Lemma 3.2. Let S ⊂ N a semigroup. If S has the property (C), then S̃ = 𝔫.

Proof. Since S̃ is arcwise connected the subgroup G(S̃) of N generated by S̃ in N is also arcwise connected. It follows that the group G(S̃) generated by S̃ is an analytic subgroup of N (see [1], Theorem V.1.1). Therefore the Lie algebra of G(S̃) is the subalgebra ⟨⟨S⟩⟩ of 𝔫 generated by S. Since S has the property (C) and the maximal semigroups with interior points in N are in oneto-one correspondence with the half-spaces in 𝔫 whose boundary is a subalgebra, (cf. [1], Corollary V.5.41), we have that ⟨⟨S⟩⟩ = 𝔫. Therefore, G(S̃) = N. This implies that S̃ has non void dense interior in N, (see [1], Theorem V.1.16). Since S ⊂ S̃ we conclude that S̃ = N.

The next result is purely combinatorial geometry, but we are unaware of a direct reference in the literature.

Proposition 3.3. Let Γ a lattice of N and S ⊂ Div(Γ) a semigroup. If S has the property (C), then there exists I ⊂ S finite with property (C).

Proof. The maximal semigroups with interior points in N are the closed half spaces in 𝔫 bounded by a hyperplane containing the derived algebra 𝔫′ = [𝔫, 𝔫], (see [1], Corollary V.5.41). Since every proper subsemigroup with interior points in a topological group is contained in a maximal subsemigroup with interior points it is sufficient to consider the maximal semigroups with non-empty interior.

Let φ_{1}, φ_{2}, …, φ_{l} be a system of generators of the annihilator [𝔫, 𝔫]^{⊥} ⊂ 𝔫^{∗} of [𝔫, 𝔫]. For 1 ≤ i ≤ l and j ∈ {0, 1} let

V_{j}^{i} = {x ∈ 𝔫 : (−1)^{j}φ_{i}(x) > 0}

be the open half-spaces given by the linear functional φ_{i}. For σ = {j_{1}, j_{2}, …, j_{l}} such that j_{k} ∈ {0, 1} we consider also the open cones

V_{σ} = ⋂_{i=1}^{l} V_{j_{i}}^{i}.

By the above lemma we have that S̃ = 𝔫. Hence for each subset σ there are a_{1}, a_{2}, …, a_{p} ∈ S and non-negative numbers t_{1}, t_{2}, …, t_{p} such that

x_{σ} = t_{1}a_{1} ∗ t_{2}a_{2} ∗ ⋯ ∗ t_{p}a_{p} ∈ V_{σ}.

Since ℚ is dense in ℝ we can take t_{1}, t_{2}, …, t_{N} ∈ ℚ. Let d ∈ ℕ be such that dt_{i} ∈ ℕ, i = 1, …, p, and consider the element

a_{σ} = dt_{1}a_{1} ∗ dt_{2}a_{2} ∗ ⋯ ∗ dt_{p}a_{p}

in S. We claim that a_{σ} ∈ V_{σ}. In fact, using the Campbell-Hausdorff formula for 𝔫, we get explicitly that

a_{σ} = ∑_{i=1}^{p} dt_{i}a_{i} + Y and x_{σ} = ∑_{i=1}^{p} t_{i}a_{i} + Z

with Y, Z ∈ [𝔫, 𝔫]. Therefore, if 1 ≤ i ≤ l we have that

0 < (−1)^{j_{i}}φ_{j_{i}}(x_{σ}) = (−1)^{j_{i}}φ_{j_{i}}(∑_{k=1}^{p} t_{k}a_{k}),

and, then

0 < (−1)^{j_{i}}φ_{j_{i}}(∑_{k=1}^{p} dt_{k}a_{k}) = (−1)^{j_{i}}φ_{j_{i}}(a_{σ}),

Therefore a_{σ} ∈ V_{j_{i}}^{i} ∩ S for i = 1, 2, …, l and then a_{σ} ∈ V_{σ} ∩ S.

Now let M be a maximal semigroup with interior points in N. We can assume without loss of generality that

M = {x ∈ 𝔫 : λ(x) ≥ 0},

where λ is a non zero linear functional which is identically zero in [𝔫, 𝔫]. Let us suppose that

λ = b_{1}φ_{1} + b_{2}φ_{2} + ⋯ + b_{l}φ_{l}

and let σ = (j_{1}, j_{2}, ..., j_{l}) be such that

j_{i} = \{ \begin{matrix}0 & if & b_{i} < 0 \\ 1 & if & b_{i} ≥ 0. \end{matrix}

Put a_{σ} ∈ V_{σ} ∩ S. Then b_{i}φ_{i}(a_{σ}) ≤ 0 for 1 ≤ i ≤ l and λ(a_{σ}) < 0 because φ_{i}(a_{σ}) ≠ 0 and λ ≠ o. This shows that S is not contained in M.

Let I be the subset of elements a_{σ} where σ = (j_{1}, j_{2}, …, j_{l}) and j_{i} ∈ {0, 1}. Then I is a finite subset of S that property (C).

Theorem 3.4. Let N be a connected and simply connected nilpotent Lie group, Γ ⊂ G a lattice and S ⊂ Div(Γ) a semigroup with property C. Then S is a group.

Proof. Pick a ∈ S. By the previous proposition there are a_{1}, a_{2}, …, a_{l} in S that have property (C). Let Ω be the semigroup generated by a_{1}, a_{2}, …, a_{l} and a. By Proposition 2.6, Ω is a discrete semigroup of N. Therefore Ω satisfies the conditions of Theorem 4.1 in [7]. Hence Ω is a subgroup of N. This implies that a^{−1} ∈ S, showing that S is a group.

From this theorem we can get the maximal semigroups in a rational group. The proof of the next result is similar to that of Corollary 4.3 in [6].

Corollary 3.5. Let Γ a lattice in a connected and simply connected nilpotent Lie group N. If S is a maximal subsemigroup of Div(Γ), then S = Div(Γ) ∩ T , where T is a maximal subsemigroup of N with non-void interior. Moreover, any semigroup of Div(Γ), which is not a group, is contained in a maximal one.

4. The non-simply connected case

Let N be a connected nilpotent Lie group, π : Ñ ⟶ N be the universal covering of N and put D = ker(π), a discrete central subgroup of Ñ. If S is a semigroup of N with property (C) such that the identity of N is in S and \overline{S} is a proper semigroup of Ñ, with non-void interior, which contains S̃ = π^{−1}(S) then D ⊂ S̃ ⊂ \overline{S} and π(\overline{S}) is a proper semigroup of N with non-void interior which contains S. Therfore π^{−1}(S) also has the property (C).

Now, if Γ is a lattice in N, then Γ̃ = π^{−1}(Γ) is a lattice in Ñ and D ⊂ Γ̃.

Corollary 4.1. Let N be a connected nilpotent Lie group, Γ ⊂ N a lattice. Let D be a divisible subgroup of N that contains Γ with D ⊂ π(Div(Γ̃). If S ⊂ D is a semigroup which has the property (C) then S is a subgroup.

Proof. Assume without loss of generality that S contains the identity of N. Since D ⊂ π(Div(Γ̃), π^{−1}(D) ⊂ Div(Γ̃) and thus S̃ = π^{−1}(S) ⊂ Div(Γ̃). But S̃ has property (C). By Theorem 3.4, S̃ is a group. Therefore S = π(S̃) is a group.

References

[1] Hilgert, J., K. H. Hofmann, and J. D. Lawson, “Lie Groups, Convex Cones, and Semigroups”, Oxford University Press, 1989.

[2] Lawson, J., Maximal semigroups of Lie groups that are total, Proc, Edinburgh Math. Soc. 30 (1987), 479–501.

[3] Margulis, G. A., “Discrete Subgroups of Semisimple Lie Groups,” Springer-Verlag, 1989.

[4] Matsushima, Y.,On the discrete subgroups and homogeneous spaces of nilpotent Lie groups, Nagoya Math. J. 2 (1951), 95–110.

[5] Raghunathan, M. S., “Discrete Subgroups of Lie Groups,” Springer-Verlag, 1972.

[6] Rocio, O. G., and L. A. B. San Martin, Discrete semigroups in nilpotent Lie groups, Semigroup Forum 51 (1995), 125–133.

[7] Rocio, O. G., and L. A. B. San Martin, Semigroups in lattice of solvable Lie groups, Journal of Lie Theory 5 (1996), 179–202.

Osvaldo Germano do Rocio Departamento de Matemática Universidade Estadual de Maringá Cx.Postal 331 87.020-900 Maringá-Pr Brasil

Received March 5, 1998 and in final form April 10, 2000