ON DISCONTINUOUS GROUPS ACTING ON HOMOGENEOUS

SPACES WITH NONCOMPACT ISOTROPY SUBGROUPS

Toshiyuki KOBAYASHI*

Institute for Advanced Study

and

University of Tokyo

August, 1992

Abstract. Let G be a Lie group and H a closed subgroup. The action of a discrete subgroup Γ of G on G/H is not always properly discontinuous if H is non-compact. If the action of Γ is properly discontinuous, then Γ is called a discontinuous group acting on G/H. If G/H is of reductive type, it is known that there are no infinite discontinuous groups acting on G/H (called Calabi-Markus phenomenon) iff ℝ-rank G = ℝ-rank H. For more understanding of discontinuous groups we are thus interested in G/H (i) where G/H is non-reductive, and (ii) where G/H is of reductive type with ℝ-rank G = ℝ-rank H + 1. In this paper we consider Calabi-Markus phenomenon in the solvable cases for (i). We also study discontinuous groups of reductive group manifolds for (ii) and generalize a result of Kulkarni-Raymond in higher dimensions.

Keywords: locally homogeneous manifolds, Calabi-Markus phenomenon, discontinuous groups

1991 MSC 22E40

0. Introduction

One of the basic problems in geometry has been to study how local geometric struc-

ture affects the global nature of a manifold. Our concern in this paper is with a special

*The author is supported by the NSF grant DMS-9100383.

Typeset by AMS-TEX

problem of this kind: “What is a possible fundamental group π_{1} of a manifold which is locally isomorphic to a particular homogeneous space ?” This is similar to a well-studied problem in differential geometry about a possible fundamental group π_{1} of a manifold under some curvature conditions. Here are some typical examples:

1) In the physics of relativistic cosmology, the space-time continuum is taken to be a Lorentz manifold M^{4}. Here a Lorentz manifold M^{n} is an n-dimensional manifold which bears a pseudo-Riemannian metric of type (n − 1, 1). The manifold M is said to be complete if every geodesic can be defined on all time intervals. A relativistic spherical space form is a complete Lorentz manifold M^{n} for n ≥ 3 with constant curvature K = +1. It is a remarkable result due to Calabi-Markus that any relativistic space form is noncompact and has a finite fundamental group π_{1} ([C-M]).

2) A Clifford-Klein form of a connected and simply connected Riemannian manifold M is a Riemannian manifold whose universal Riemannian covering is isomorphic to M. For example, any compact Riemann surface of genus ≥ 2 is regarded as a compact Clifford-Klein form of the Poincaré plane. More generally, there always exists a compact Clifford-Klein space form of a Riemannian symmetric space of the noncompact type ([Bo], [Bo-HC], [M-T]).

3) An affine manifold M^{n} is a manifold which admits a torsion free affine connection whose curvature tensor vanishes. It is called Auslander’s conjecture that the fundamental group π_{1} of any compact complete affine manifold is virtually solvable (see [A], [Mi], [Ma] for instance). These cases can be reformulated in the context of discontinuous groups acting on

homogeneous spaces as follows. Let G be a Lie group and H a closed subgroup of G. A subgroup Γ of G is called a discontinuous group acting on a homogeneous space G/H if the action of Γ on G/H from the left is properly discontinuous. A discontinuous group acting on G/H is automatically discrete in G, whatever H is. A distinguishing feature in our setting is that H is noncompact, and consequently, a discrete subgroup is not necessarily a discontinuous group acting on G/H. This is the primary difficulty in our study. On the other hand, in the above definition of a discontinuous group we do not require the freeness of the action. A small price to pay is that the double coset space Γ\G/H is not necessary a manifold but only a V -manifold in the sense of Satake [Sa]. However, if there exists a cocompact discontinuous group Γ acting on G/H (i.e. a discontinuous group acting on G/H such that Γ\G/H is compact), then we can replace Γ by a subgroup Γ′ of finite index in Γ so that Γ′\G/H is a compact smooth manifold by virtue of the result in [Se]. Now the above examples are reformulated respectively as follows: 1′) Any discontinuous group acting on SO(n, 1)/SO(n − 1, 1) is finite. 2′) There exists a cocompact discontinuous group acting on G/K if G is a real linear

semisimple Lie group and if K is a maximal compact group of G. 3′) Any cocompact discontinuous group acting on GL(n, ℝ) ⋉ ℝ^{n}/GL(n, ℝ) is conjectu-

ally virtually solvable.

Here are some comments on recent progress on (1′), (2′) and (3′). The conjecture (3′) remains open except for some special cases such as O(n)⋉ℝ^{n}/O(n) (Bieberbach’s theorem, see [R], Corollary 8.26), O(n, 1) ⋉ ℝ^{n}/O(n, 1) [G-K], G ⋉ ℝ^{n}/G where G is a subgroup of GL(n, ℝ) which is locally isomorphic to a direct product of semisimple Lie groups of rank 1 [To]. It also remains open to classify the homogeneous spaces of reductive type (see §3 for definition) that admit compact Clifford-Klein space forms (see (2′)). Partial results have been obtained in [Bo; M-T; Ku; Ko1; Ko3]. The feature in (1′) without infinite discontinuous groups is called Calabi-Markus phenomenon. In a previous paper we have established a criterion for Calabi-Markus phenomenon in the case of a homogeneous space of reductive type:

Fact 0.1 (see [C-M; Wo1; Wo2; Wo3; Ku; Ko1]). Let G/H be a homogeneous space of reductive type (see §3 for definition). Then the following conditions are equivalent:

(1) Any discontinuous group acting on G/H is finite.

(2) ℝ-rank G = ℝ-rank H.

In view of this, we wish to proceed a further step by posing the following questions:

Question 0.2. Suppose G/H is not of reductive type. Find a condition that G/H admits an infinite discontinuous group.

Question 0.3. Suppose G/H is of reductive type with ℝ-rank G − ℝ-rank H = 1. What can we say about a possible infinite discontinuous group acting on G/H ?

In answer to Question (0.2) for solvable homogeneous spaces, we shall prove

Theorem 1 (see §2). Suppose G is a solvable Lie group and H is a proper closed subgroup of G. Then there exists a discrete subgroup Γ of G acting on G/H properly discontinuously and freely such that the fundamental group π_{1}(Γ\G/H) is infinite.

This result is in sharp contrast to reductive case; For example, the following homogeneous spaces G/H = GL(n, ℂ)/GL(n, ℝ), GL(m + n, ℝ)/GL(m, ℝ) × GL(n, ℝ), U(p, q)/SO(p, q) which are of reductive type, do not admit infinite discontinuous groups by Fact (0.1).

Given a subgroup Φ of G and a homomorphism ρ: Φ → G, we form a subgroup of G × G as

Φ(ρ) := {(γ, ρ(γ)) : γ ∈ Φ} (⊂ G × G).

If the homomorphism ρ is the trivial representation 1, then the action of Φ(1) = Φ × 1 on G ≃ G×G/ diag G is nothing but the one from the left. In this sense we might regard the action of Φ(ρ) as a ‘deformation’ of the left action of Φ. If Φ is a discrete subgroup of G and if the image ρ(Φ) is relatively compact, then Φ(ρ) is also a discontinuous group acting on the group manifold G × G/ diag G.

For example, suppose that Φ ⊂ P SL(2, ℝ) is the fundamental group of a compact Riemann surface M of genus g (≥ 2), and fix generators of the first homology group H_{1}(M, ℤ) ≃ ℤ^{2g}. Then we find the moduli space of group homomorphisms from Φ to SO(2) to be Hom(Φ, SO(2)) ≃ 𝕋^{2g}. That is, λ ∈ 𝕋^{2g} defines a homomorphism φ_{λ}: ℤ^{2g} → 𝕋 ≃ SO(2) ⊂ P SL(2, ℝ), and we get a homomorphism ρ_{λ}: Φ → P SL(2, ℝ) as a composition of φ_{λ} and Φ ≃ π_{1}(M) → Φ/[Φ, Φ] ≃ H_{1}(M, ℤ) ≃ ℤ^{2g}. Then Φ(ρ_{λ}) = {(γ, ρ_{λ}(γ)) : γ ∈ Φ} forms a family of cocompact discontinuous groups acting on the group manifold of G × G/ diag G parametrized by λ ∈ 𝕋^{2g}.

Even though it is hopeless to classify all discontinuous groups arising in Question (0.3) because it involves all discrete subgroups of a semisimple Lie group G with ℝ-rank G = 1 (i.e. discontinuous groups acting on G/{e}), we can describe some aspects of the structure of such a discontinuous group when G/H is a group manifold G′ × G′/ diag G′, where ℝ-rank(G′ × G′) − ℝ-rank(G′) = 1 (i.e. ℝ-rank G′ = 1).

Theorem 2 (see Corollary 3.4). Let G be a connected noncompact reductive linear group. Then the following conditions are equivalent.

(1) ℝ-rank G = 1

(2) For any torsionless discontinuous group Γ acting on G × G/ diag G, we can find a subgroup Φ ⊂ G and a homomorphism ρ: Φ → G such that Γ = {(γ, ρ(γ)) : γ ∈ Φ} up to switch of factor.

Remark 1. R.Kulkarni and F.Raymond first proved (1) ⇒ (2) when G = SL(2, ℝ) in their study of three dimensional Lorentz space-forms (see Theorem 5.2 and Introduction in [Ku-R]). Their proof depends on the key lemma that no discontinuous group acting on G × G/ diag G contains an abelian subgroup ≃ ℤ^{2} if G = SL(2, ℝ). However, this is not always true even if we assume G is of ℝ-rank 1. For example, we can show that there exists an abelian discontinuous group ≃ ℤ^{n−1} acting on G × G/ diag G if G = SO(n, 1).

Remark 2. Theorem 2 leads us to a natural question about the condition on the pair Φ and ρ such that Φ(ρ) is a discontinuous group acting on the group manifold G × G/ diag G. In the case G = SL(2, ℝ), it is known to be necessary that Φ is discrete (possibly after a switch of factor) [Ku-R]. It is not known to the author whether it is necessary that Φ is discrete (after a switch of factor) for a general ℝ-rank 1 group. On the other hand, it is sufficient for the discontinuity of Φ(ρ) that Φ is discrete and ρ has a relatively compact image. There are a number of examples of such homomorphisms ρ. For instance, if G is complex semisimple Lie group and Φ is arithmetic, then we can find a nontrivial homomorphism ρ into a maximal compact group of G (e.g. [Z], Example 5.2.12). If G = SO_{0}(n, 1) and Φ is an arithmetic cocompact discrete subgroup of G such that the first Betti number b_{1} of Φ\SO_{0}(n, 1)/SO(n) does not vanish (Thurston’s conjecture, see [L]), then we have a continuous family of discontinuous groups Ψ(ρ) parametrized by ρ ∈ Hom(Φ, 𝕋^{[ \frac{n}{2} ]}) ≃ 𝕋^{b_{1}[ \frac{n}{2} ]} when G = SO_{0}(n, 1), as we saw for G = P SL(2, ℝ). Finally we also remark that in the case G = SL(2, ℝ), some other sufficient conditions for (Φ, ρ) are also known that assure the discontinuity of Φ(ρ) on G × G/ diag G (see [G]), but it still remains open to classify all possible (Φ, ρ) such that Φ(ρ) is a cocompact discontinuous group acting on SL(2, ℝ) × SL(2, ℝ)/ diag SL(2, ℝ).

Remark 3. It is remarkable that the example in Remark 2 shows that “local rigidity” fails in higher dimensions in the case where the isotropy group is not compact. To be more precise, let Φ be a finitely generated group and G a Lie group. Let R(Φ, G) be the set of all homomorphisms of Φ into G equipped with the topology of pointwise convergence. Let H be a closed subgroup of G. We define

R(Φ, G, H) := {u ∈ R(Φ, G) : u is injective, u(Φ) is a discontinuous group acting on G/H} .

R_{0}(Φ, G, H) := {u ∈ R(Φ, G, H) : u(Φ)\G/H is compact} .

A homomorphism u ∈ R(Φ, G, H) is called locally rigid if the orbit of u in R(Φ, G, H) under G is open in R(Φ, G, H). If G is semisimple with trivial center and no compact factors, then local rigidity holds for any u ∈ R_{0}(Φ, G, {e}) (or u ∈ R_{0}(Φ, G, K) where K is a maximal compact group of G) unless G = P SL(2, ℝ) (Weil’s rigidity theorem). However, in the case of G = SO_{0}(n, 1), local rigidity fails for G × G/ diag G because two generic elements in \{Φ(ρ) : ρ ∈ Hom(Φ, 𝕋^{b_{1}[ \frac{n}{2} ]})\} ⊂ R_{0}(Φ, G × G, diag G) (with the notation of Remark 2) are not conjugate under G × G.

1. Preliminary results on proper actions

First of all, let us recall the definition of a proper continuous map.

Definition 1.1 (see [Bou]). Let f: X → Y be a continuous map between locally compact Hausdorff spaces. f is called proper iff one of the following equivalent conditions holds.

(1) f is a closed map, and f^{−1}(y) is compact for any y ∈ Y.

(2) For any topological space Z, f: X × Z → Y × Z is a closed map.

(3) f^{−1}(S) is compact for any compact subset S of Y.

If f is a proper map, then it follows easily that a closed subset Z of X is compact iff f(Z) is contained in some compact set of Y.

Definition 1.2. Suppose that a locally compact topological (Hausdorff) group G acts continuously on a locally compact Hausdorff space X. This action is called proper iff the map G × X ∋ (g, x) ↦ (x, g · x) ∈ X × X is proper. Equivalently, G_{S} := {g ∈ G : g · S ∩ S ≠ ∅} is compact for every compact subset S in X. The action is called properly discontinuous iff G is discrete and acts properly on X.

The following elementary lemma deals with proper actions under a equivariant map.

Lemma 1.3. Let G_{i} (i = 1, 2) be locally compact groups and L_{i}, H_{i} ⊂ G_{i} be closed subgroups. Suppose that f: G_{1} → G_{2} is a (continuous) homomorphism such that f(L_{1}) ⊂ L_{2}, f(H_{1}) ⊂ H_{2}. Assume that f(L_{1}) is closed in G_{2}.

1) Assume that L_{1} ∩ Ker f is compact. If the L_{2} action on G_{2}/H_{2} is proper, then the L_{1} action on G_{1}/H_{1} is also proper.

2) Assume that f(G_{1})H_{2} = G_{2}, that G_{1} → G_{2}/H_{2} is an open map, and that the quotients L_{2}/f(L_{1}), f^{−1}(H_{2})/H_{1} are compact. If the L_{1} action on G_{1}/H_{1} is proper, then the L_{2} action on G_{2}/H_{2} is also proper.

Remark 1.4. If G_{i} are (separable) Lie groups, then the first assumption f(G_{1})H_{2} = G_{2} in (2) implies the second one that the map G_{1} → G_{2}/H_{2} is open.

Remark 1.5. In (2), the assumption f(G_{1})H_{2} = G_{2} looks very strong. However, we cannot replace this assumption by the weaker one that G_{2}/f(G_{1}) is compact. For example, let G_{1} = ℝ^{n} and W be a finite subgroup of GL(n, ℝ). Then we form a semidirect product G_{2} := W ⋉ ℝ^{n}. Let f: G_{1} ↪ G_{2} be a natural inclusion. Fix two abelian subspaces L_{1}, H_{1} ⊂ G_{1} = ℝ^{n} such that L_{1} ∩ H_{1} = {0} and that w · L_{1} ∩ H_{1} ≠ {0} for some w ∈ W. Define subgroups of G_{2} by L_{2} := L_{1}, H_{2} := H_{1}, where we regard G_{1} ⊂ G_{2}. Then L_{1} acts properly on G_{1}/H_{1}, while L_{2} does not act properly on G_{2}/H_{2}. The situation of this kind turns up as a reduction of the case where G_{i}, L_{i}, H_{i} are connected reductive groups (see [Ko1], Theorem 4.1).

Proof of Lemma (1.3). 1) Fix any compact subset S of G_{1}. We want to show the set {g ∈ L_{1} : (g · S mod H_{1}) ∩ (S mod H_{1}) ≠ ∅ in G_{1}/H_{1}} = L_{1} ∩ SH_{1}S^{−1} is compact. In view of

f(L_{1} ∩ SH_{1}S^{−1}) ⊂ L_{2} ∩ f(S)H_{2}f(S)^{−1},

f(L_{1} ∩ SH_{1}S^{−1}) is contained in a compact set if L_{2} acts on G_{2}/H_{2} properly. Then L_{1}∩SH_{1}S^{−1} is compact, since f_{|L_{1}} : L_{1} → L_{2} is a proper map because it is a composition of proper maps: L_{1} → L_{1}/L_{1} ∩ Ker f ≃ f(L_{1}) ↪ L_{2}. That is, L_{1} acts on G_{1}/H_{1} properly.

2) As f(L_{1}) is a closed and cocompact subgroup of L_{2}, L_{2} acts properly iff f(L_{1}) (⊂ L_{2}) acts properly. So we may and do assume f(L_{1}) = L_{2}. Take a compact set S_{1} of G_{1} such that f^{−1}(H_{2}) = S_{1}H_{1}. We may assume that S_{1} contains the unit of G_{1}. Fix any compact subset S of G_{2}. Let us show that L_{2} ∩ SH_{2}S^{−1} is compact. The existence of a compact subset S̃ of G_{1} such that f(S̃)H_{2} ⊃ S follows from the fact that G_{1}/f^{−1}(H_{2}) is homeomorphic to G_{2}/H_{2} (see the assumptions that G_{1} → G_{2}/H_{2} is an open map and f(G_{1})H_{2} = G_{2}). Then we have

f^{−1}(L_{2} ∩ SH_{2}S^{−1}) ⊂ f^{−1}(L_{2}) ∩ S̃f^{−1}(H_{2})S̃^{−1}.

In particular, (f_{|L_{1}})^{−1} (L_{2}∩SH_{2}S^{−1}) is compact if L_{1} acts properly on G_{1}/H_{1}, because

(f_{|L_{1}})^{−1} (L_{2} ∩ SH_{2}S^{−1}) ⊂ L_{1} ∩ f^{−1}(L_{2}) ∩ S̃f^{−1}(H_{2})S̃^{−1} ⊂ L_{1} ∩ S̃S_{1}H_{1}S_{1}^{−1}S̃^{−1}.

Under our assumption f(L_{1}) = L_{2}, we have L_{2} ∩ SH_{2}S^{−1} = (f_{|L_{1}}) ∘ (f_{|L_{1}})^{−1} (L_{2} ∩ SH_{2}S^{−1}) is compact. Thus L_{2} acts on G_{2}/H_{2} properly. □

2. Homogeneous spaces of solvable groups

First we recall a nice topological property of a subgroup of a solvable Lie group due to Chevalley.

Fact 2.1 [Ch]. Let G be a 1-connected (real) solvable Lie group and H be a connected subgroup of G. Then H is closed and 1-connected.

Our main theorem in this section is,

Theorem 2.2. Let G be a solvable Lie group and H a proper closed subgroup of G. Then there exists a discrete subgroup Γ of G that acts on G/H properly discontinuously and freely such that the fundamental group π_{1}(Γ\G/H) is infinite.

If ♯π_{1}(G/H) = ∞, then we can take Γ = {e} and we are done. Hereafter we suppose π_{1}(G/H) is a finite group. We put G_{2} := G, H_{2} := H, G_{1} := the universal covering group of G_{2} and H_{1} := the connected subgroup of G_{1} with the Lie algebra 𝔥. We write f: G_{1} → G_{2} for the covering map. Because π_{1}(G/H) = π_{1}(G_{2}/H_{2}) = π_{1}(G_{1}/f^{−1}(H_{2})) = f^{−1}(H_{2})/H_{1}, and because π_{1}(G/H) is a finite group, we can apply Lemma (1.3)(2) with any subgroup L_{1} ⊂ G_{1} and with L_{2} := f(L_{1}). Therefore, in order to prove Theorem (2.2) it suffices to prove:

Theorem 2.2′. Let G be a 1-connected (real) solvable group and H be a connected proper subgroup of G. Then there exists a discontinuous group acting on G/H which is isomorphic to ℤ.

Proof. We proceed by the induction on the dimension of G. Theorem (2.2)′ is clear when dim G = 1, namely, when G ≃ ℝ ⊃ H ≃ {0}. Suppose that dim G ≥ 2. Then there exists a connected normal subgroup N of G with 0 < dim N < dim G. We will divide into two cases according as HN ⫋ G or HN = G.

I) Assume that HN ⫋ G. The subgroup HN is connected and therefore closed by Fact (2.1). So \overline{H} := H/H ∩ N = HN/N is a proper closed subgroup of \overline{G} := G/N. We write the canonical projection π: G → \overline{G} = G/N. It follows from the inductive assumption that we can find a discrete subgroup \overline{Γ} of \overline{G} such that \overline{Γ} is isomorphic to ℤ and acts on \overline{G}/\overline{H} properly. Fix an element γ ∈ G such that π(γ) is a generator of \overline{Γ}. Put Γ := ⟨γ⟩. We have π(Γ) = \overline{Γ}, and therefore Γ ≃ ℤ and Γ ∩ N = \{e\}. On the other hand, \overline{Γ} is discrete and so is Γ. Applying Lemma (1.3)(1), we have now shown that Γ acts on G/H properly discontinuously.

II) Assume that HN = G. We have G/H ≃ N/N ∩ H and N ∩ H ⫋ N. Since π_{1}(N/N ∩ H) = π_{1}(G/H) = {e}, N ∩ H is connected. Thus (N, N ∩ H) satisfies the assumption of Theorem (2.2)′ and dim N < dim G. Therefore we can find a discrete group Γ ≃ ℤ of N which acts on N/N ∩ H from the inductive assumption. Clearly, Γ is a subgroup of G acting properly discontinuously on G/H. □

3. ℝ-rank 1 semisimple group manifolds

Throughout this section, we assume that G is a connected real reductive linear Lie group. First we set up notation. Let G be a real linear reductive Lie group, with real Lie algebra 𝔤. Given a Cartan involution θ of G, we write a Cartan decomposition of its Lie algebra as 𝔤 = 𝔨 + 𝔭. Fix a maximally abelian subspace 𝔞 ⊂ 𝔭. 𝔞 is called a maximally split abelian subspace for G. We write W (𝔤, 𝔞) for the Weyl group associated to the root system of Σ(𝔤, 𝔞). Let ℝ-rank G := dim 𝔞, the real rank of G. Let H be a closed subgroup of G which has finitely many connected components. If there exists a Cartan involution of G which stabilizes H, then H is called reductive in G and G/H is called a homogeneous space of reductive type. In this case, H has a Cartan decomposition H = (H ∩ K) exp(𝔥 ∩ 𝔭), and 𝔥 is reductive in 𝔤, namely, the adjoint representation 𝔥 → 𝔤𝔩(𝔤) is completely reducible. Let 𝔞_{H} be a maximally split abelian subspace for H. Then there exists an element g of G such that Ad(g)𝔞_{H} ⊂ 𝔞. Put 𝔞(H) := Ad(g)𝔞_{H}, which is uniquely defined up to conjugacy of W (𝔤, 𝔞).

We shall find some structure theorem of a discontinuous group acting on a group manifold G × G/ diag G when ℝ-rank G = 1.

Lemma 3.1. If ℝ-rank G = 1 and x ∈ G is a semisimple and non-elliptic element, then L := Z_{G}(x) is a direct product of a compact group and ℝ.

Proof. There exists a Cartan involution θ of G such that θL = L. Then we have a Cartan decomposition of L:

φ: (𝔩 ∩ 𝔭) × (L ∩ K) →^{∼} L, (X, k) ↦ (exp X)k.

Let us denote by C the center of L. According to the above decomposition, we have C = exp(𝔠 ∩ 𝔭)(C ∩ K). It follows from the assumption that ⟨x⟩ := {x^{n} : n ∈ ℤ} ≃ ℤ is a discrete subgroup of G and ⟨x⟩ ⊂ C. Since G is a linear group, C ∩ K is compact and so C is a closed abelian group with at most finitely many connected components. Therefore dim 𝔠∩𝔭 ≥ 1. On the other hand, 1 = ℝ-rank G ≥ ℝ-rank L = ℝ-rank[L, L]+ dim 𝔠 ∩ 𝔭. Thus we have ℝ-rank[L, L] = 0 and therefore 𝔩 ∩ 𝔭 = 𝔠 ∩ 𝔭 (≃ ℝ). Hence φ: (𝔠 ∩ 𝔭) × (L ∩ K) → L is a Lie group isomorphism, where we regard 𝔠 ∩ 𝔭 as an additive group. □

Lemma 3.2. If ℝ-rank G = 1 and Γ is an infinite discrete subgroup of G, then there exists a compact set S of G such that SΓS^{−1} = G.

Proof. It is known that any infinite discrete subgroup Γ in a linear Lie group contains an element of infinite order. Fix such an element γ ∈ Γ. In order to prove Lemma (3.2) it suffices to show the existence of a compact set S such that S⟨γ⟩S^{−1} = G. Let γ = γ_{s}γ_{u} be its Jordan decomposition (see [War] Proposition 1.4.3.3), where γ_{s} is semisimple and γ_{u} is unipotent such that γ_{s}γ_{u} = γ_{u}γ_{s}. We divide into two cases according as γ_{s} is elliptic or not.

I) Assume that γ_{s} is a non-elliptic element of G. It follows from Lemma (3.1) that the only unipotent element of Z_{G}(γ_{s}) is the identity. Since γ_{u} ∈ Z_{G}(γ_{s}), we have γ_{u} = 1. Thus γ = γ_{s} is contained in a maximally split Cartan subgroup J. Choose a Cartan involution θ which stabilizes J. We write the corresponding Cartan decomposition G = exp 𝔭K and we write J = T A, where T := J ∩K and A := J ∩exp 𝔭. We can write γ = t exp(Y) where t ∈ T, Y ∈ 𝔞. Define a compact subset of G by S := K {exp sY : 0 ≤ s ≤ 1}. Then S⟨γ⟩S^{−1} ⊃ KAK = G.

II) Assume that γ_{s} is elliptic. Then γ_{u} ≠ 1 since ⟨γ⟩ = {γ_{s}^{n}γ_{u}^{n} : n ∈ ℤ} is discrete in G. By the theorem of Jacobson-Morozov, there is a Lie group homomorphism ψ: SL(2, ℝ) → G such that ψ (( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )) = γ_{u}. There is a Cartan involution θ of G such that θψ(SL(2, ℝ)) = ψ(SL(2, ℝ)) (see [He], p.277). In particular, A := ψ (\{( \begin{matrix}a & 0 \\ 0 & a^{−1} \end{matrix} ) : a > 0\}) is a maximally split abelian subgroup of G, which is of

ℝ-rank 1. Define a compact subset of G by S := Kψ (\{( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) : 0 ≤ x ≤ 1\}) \overline{⟨γ_{s}⟩}.

Then

S⟨γ⟩S^{−1} ⊃ Kψ (\{( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) : x ∈ ℝ\}) K ⊃ Kψ (SL(2, ℝ)) K ⊃ KAK = G.

□

Theorem 3.3. Let G be a connected reductive linear Lie group. Then the following conditions are equivalent.

(1) ℝ-rank G ≥ 2

(2) There exist infinite discrete subgroups Γ_{i} of G (i = 1, 2) such that Γ := Γ_{1} ×Γ_{2} acts properly discontinuously on the group manifold G × G/ diag G.

Proof. We may restrict ourselves to the case where G is noncompact, namely, where ℝ-rank G ≥ 1.

Suppose that ℝ-rank G ≥ 2. We can find abelian subspaces 𝔞_{1}, 𝔞_{2} ⊂ 𝔞 such that dim 𝔞_{i} ≥ 1 and that W (𝔤, 𝔞) · 𝔞_{1} ∩ 𝔞_{2} = {0}. Put A_{i} := exp 𝔞_{i}, then A_{1} acts properly on G/A_{2} from [Ko1], Theorem 4.1. Take any lattices Γ_{i} in abelian Lie groups A_{i} (i = 1, 2). Then Γ_{1} acts properly discontinuously on G/Γ_{2}, or equivalently, the discrete group Γ_{1} × Γ_{2} acts properly discontinuously on G × G/ diag G.

Conversely, suppose that ℝ-rank G = 1. We recall that a subgroup Γ of G × G acts properly on G × G/ diag G iff CΓC^{−1} ∩ diag G is compact for any compact subset C of G×G. In particular, if there exists a compact set C in G×G such that CΓC^{−1} = G×G, then Γ acts on G × G/ diag G properly only if G is compact. Let Γ_{i} (i = 1, 2) be both infinite discrete subgroups of G. It follows from Lemma (3.2) that there exists a compact set S of G such that SΓ_{i}S^{−1} = G. In particular, (S ×S)(Γ_{1} ×Γ_{2})(S^{−1} ×S^{−1}) = G×G. Therefore the action of Γ_{1} × Γ_{2} on G × G/ diag G is not properly discontinuous because G is noncompact. □

Corollary 3.4. Let G be a connected noncompact reductive linear group. Then the following conditions are equivalent.

(1) ℝ-rank G = 1

(2) Any torsionless discontinuous group Γ in G×G/ diag G is of the following form up to switch of factor: Γ = {(γ, ρ(γ)) : γ ∈ Φ}, with Φ ⊂ G a subgroup and with ρ: Φ → G a homomorphism.

Proof. 2) ⇒ 1) If ℝ-rank G ≥ 2, then there exist discrete subgroups Γ_{i} ≃ ℤ^{n_{i}} (n_{i} ≥ 1) of G such that Γ_{1} × Γ_{2} acts properly discontinuously on G × G/ diag G from Theorem (3.3).

1) ⇒ 2) Suppose that Γ is a torsion free discontinuous group acting on G×G/ diag G. Let p_{j}: G × G → G (j = 1, 2) be natural projections to the j-th factor. Let Γ_{j} := Ker p_{j} ∩ Γ for j = 1, 2. Then Γ_{1} × Γ_{2} is regarded as a subgroup of Γ ⊂ G × G, and so is also a discontinuous group acting on G × G/ diag G. It follows from Theorem (3.3) that at least one of Γ_{j} must be finite if ℝ-rank G = 1. We can assume Γ_{1} is a finite group after switching factor if necessary. As Γ is torsion-free, a finite subgroup Γ_{1} must be trivial, namely, p_{1}_{|Γ} : Γ → G is injective. Now Γ is of the desired form if we define Φ := p_{1}(Γ) and ρ := p_{2} ∘ p_{1}_{|Γ}^{−1}. □

⟨ Acknowledgement ⟩

The author would like to thank Professors A.Borel, C.Conley, I.M.Gel’fand and W.M.Goldman for their comments and interest in this work.

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Institute for Advanced Study, School of Mathematics, Princeton, NJ. 08540, USA and Department of Mathematical Sciences, University of Tokyo, Meguro, Komaba, 153, Tokyo, Japan