PROPER ACTION ON A HOMOGENEOUS

SPACE OF REDUCTIVE TYPE

Toshiyuki Kobayashi^{∗}

Department of Mathematics University of Tokyo Hongo, Tokyo 113 Japan.

Abstract.

An action of L on a homogeneous space G/H is investigated where L, H ⊂ G are reductive Lie groups.

A criterion of the properness of this action is obtained in terms of the little Weyl group of G. In particular, ℝ-rankG = ℝ-rankH iff Calabi-Markus phenomenon occurs, i.e. only finite subgroups of G can act properly discontinuously on G/H. Then by using cohomological dimension theory of a discrete group, L\G/H is proved compact iff d(G) = d(L)+d(H), where d(G) denotes the dimension of a Riemannian symmetric space associated with G, etc.

These results apply to the existence problem of lattice in G/H. Several series of classical pseudo-Riemannian homogeneous spaces are found to admit non-uniform lattice as well as uniform lattice, while some necessary condition for the existence of uniform lattice is obtained when rankG = rankH.

∗Partially supported by Grant-in-Aid for Scientific Research (No.63740074).

2 TOSHIYUKI KOBAYASHI^{∗}

1. Introduction

Let H be a reductive subgroup in a real reductive linear group G. The purpose of this paper is to study a properly discontinuous action on a homogeneous space G/H of reductive type.

If H is compact, it is a famous result due to Borel and Harish-Chandra that G/H admits a uniform lattice (resp. nonuniform lattice), i.e. there is a discrete subgroup Γ in G acting properly discontinuously and freely on G/H so that Γ\G/H is compact (resp. noncompact but of finite volume) ([Bo], [B-H]). Now these theorems have been a foundation of abundant theory such as Eisenstein series in harmonic analysis on L^{2}(Γ\G/H).

However, unless H is compact, the action of a discrete group on G/H is not automatically properly discontinuous. In fact it sometimes occurs that only finite subgroup in G can act properly discontinuously on G/H. Calabi and Markus first found SO(n+1, 1)/SO(n, 1) is such a case. Now some sufficient conditions on these ‘Calabi-Markus phenomena’ have been obtained in a general case with a necessary condition in a very special case (see [C-M], [Wo], [Ku], [Wal]).

To study a properly discontinuous action on a reductive homogeneous space G/H, we will take the following approach (cf. Lemma(2.3)): Find a reductive subgroup G′ acting properly on G/H so that any discrete subgroup Γ of G′ acts automatically properly discontinuously on G/H. This approach was first partially carried out by R.S.Kulkarni ([Ku]), where he found that properties of the groups which can act properly on SO(p + 1, q)/SO(p, q) dramatically depend on the conditions of p, q by making a detailed study of a quadratic form of type (p + 1, q).

The main results of this paper will be stated in §4. We introduce it briefly: Let G/H be a homogeneous space of reductive type (Definition(2.6)). First, we give a simple criterion to tell whether the action of a reductive subgroup G′ on G/H is proper or not (Theorem(4.1)). As a corollary we show that Calabi-Markus phenomenon occurs in G/H if and only if ℝ-rank G = ℝ-rank H (Corollary(4.4)). Secondly, we obtain the necessary and sufficient condition that G′\G/H is compact under the assumption that the G′-action on G/H is proper (Theorem(4.7)). As a corollary of its proof, a certain necessary condition for the existence of uniform lattice is obtained when H has maximal rank in G (Proposition(4.10)). Finally, using these criteria in Theorem(4.1),(4.7), we find six series of non-Riemannian reductive homogeneous spaces which admit uniform lattices as well as non-uniform lattices (Proposition(4.9)).

The author expresses his sincere gratitude to Professor Toshio Oshima for his constant encouragement.

2. Notation and Preliminary Results

Let G be a topological group acting continuously on a topological space X; i.e. there is a continuous map f : G × X → X which gives a homomorphisms of G into the group of homeomorphism of X. G is said to act freely iff f(g, x) = x implies g = e for every x ∈ X; properly iff {g ∈ G : f(g, S) ∩ S ≠ ∅} is compact for every compact subset S in X; properly discontinuously iff G is discrete and acts properly on X. We shall often write g · x instead of f(g, x).

When a discrete group Γ acts smoothly on a manifold X, Γ\X is a V -manifold in the sense of [Sa] if the action is properly discontinuously; Γ\X is a manifold if it is properly discontinuously and freely. The following lemma is well-known and elementary:

Lemma(2.1). Let Γ be a group acting properly discontinuously on a locally compact Hausdorff space X. Then

1) Γ\X is also a locally compact Hausdorff space.

2) If Γ\X is compact, then Γ is finitely generated.

In fact, {γ ∈ Γ : γ · U ∩ U ≠ ∅} (U is a relatively compact open set in X such that X = Γ · U) gives a finite generator of Γ in (2).

When X is a homogeneous space G/H where H is a closed subgroup of G, our concern will be mainly restricted to the action of a subgroup of G on X = G/H via the natural left action.

Definition(2.2). Let Γ be a discrete subgroup of G. Γ is called a uniform lattice in G/H iff Γ acts on G/H freely and properly discontinuously so that Γ\G/H is compact. When G/H carries a G-invariant measure, Γ is called a lattice in G/H iff Γ acts on G/H freely and properly discontinuously so that Γ\G/H is of finite volume; a non-uniform lattice in G/H iff Γ is a lattice but not a uniform lattice. Obviously these terminologies are consistent with the usual ones when H = {e}.

Our approach of a properly discontinuous action is based on the following simple observation:

Lemma(2.3). Let a real Lie group G act on a locally compact space X and Γ be a uniform lattice in G. Then

1) The G-action on X is proper iff the Γ-action is properly discontinuous.

2) G\X is compact iff Γ\X is compact.

Proof. 1) Suppose Γ acts properly discontinuously. Take a compact subset C = C^{−1} in G so that G = C · Γ. Then for any compact subset S in X, {g ∈ G : g · S ∩ S ≠ ∅} ⊂ C · Γ_{CS}, where Γ_{CS} := {γ ∈ Γ : γ(C · S) ∩ (C · S) ≠ ∅} is a finite set. Thus the G-action is proper. The ‘only if’ part is nothing but the definition.

2) Suppose G\X is compact. We can choose finite relatively open sets U_{j} in X so that X = ⋃_{j} G · U_{j} = ⋃_{j} Γ · (C · U_{j}), showing Γ\X is compact. The converse

statement is clear. □

Suppose G is a real Lie group. We denote by G_{o} the identity component of G, by 𝔤 the Lie algebra of G, by Ad : G → GL(𝔤) the adjoint representation of G, by ad : 𝔤 → 𝔤𝔩(𝔤) the adjoint representation of 𝔤. This notation will be applied to groups denoted by other Roman letters in the same way without comment. For S ⊂ G and 𝔱 ⊂ 𝔤, N_{S}(𝔱), Z_{S}(𝔱) denote the normalizer, centralizer of 𝔱 in S respectively. Similar notations will be used when S is a subset in 𝔤.

Definition(2.4). By a real reductive linear group, we will mean a real Lie group G (not necessarily connected), contained in a connected complex reductive Lie group G_{ℂ} whose Lie algebra 𝔤_{ℂ} ≃ 𝔤 ⊗_{ℝ} ℂ.

If G is a real reductive linear group, there is a Cartan involution θ of G such that the mapping

K × 𝔭 → G, (k, X) ↦ k · exp(X)

gives a surjective diffeomorphism. Here K =^{def} {g ∈ G : θg = g} is a maximal compact subgroup of G and 𝔤 = 𝔨 + 𝔭 is the +1 and −1 eigenspace decomposition for θ (we denote by the same letter the differential of θ as usual). The ambiguity of the choice of a Cartan involution θ_{|[G,G]} is just by inner automorphisms of G (cf.

[He] Ch.VI Theorem 2.1). We set

(2.5)  d(G) := dim(G/K) = dim_{ℝ} 𝔭

Clearly this definition is independent of the choice of a Cartan involution.

We will use standard results concerning real reductive linear groups as needed when these are trivial consequences of the corresponding ones for connected cases, although the references may treat only connected semisimple Lie groups.

Definition-Lemma(2.6). Let H be a closed subgroup in a real reductive linear group G (Definition(2.4)). We call H reductive in G and G/H a homogeneous space of reductive type when the following equivalent conditions are satisfied: (2.6.1) There is a Cartan involution θ of G such that H has a polar decomposition H = (H ∩ K) · exp(𝔥 ∩ 𝔭). (2.6.2) H is of finitely many connected components and the adjoint representation ad |_{𝔥} : 𝔥 → 𝔤𝔩(𝔤) is completely reducible.

It is easy to see that H is a real reductive linear group in the sense of Definition(2.4) if H is reductive in G. Our definition of reductive subgroups are slightly stronger than the usual one by excluding the case H has infinitely many connected components.

Let H be reductive in G. Retain notations as in (2.6.1). Fix a nondegenerate invariant symmetric bilinear form on 𝔤, which we denote by ⟨ , ⟩. We can and do choose this form so that the Cartan decomposition 𝔤 = 𝔨 + 𝔭 is orthogonal for ⟨ , ⟩, and ⟨ , ⟩ is positive definite on 𝔭 and negative definite on 𝔨. This form will be restricted to 𝔭 without change of notation. Let 𝔮 be the orthogonal complement of 𝔥 in 𝔤. Then we have

Lemma(2.7). Suppose G/H is a homogeneous space of reductive type. Then we have

G = K exp(𝔮 ∩ 𝔭)H.

Furthermore, G/H is diffeomorphic to the fiber bundle

(K/H ∩ K) ×_{Ad_{|H∩K}} (𝔮 ∩ 𝔭),

with the base K/H ∩ K and fibers 𝔮 ∩ 𝔭, by the map

f : K × (𝔮 ∩ 𝔭) ∋ (k, Z) ↦ k exp(Z)H ∈ G/H.

The above lemma is well-known at least when G/H is a semisimple symmetric space. But we shall give a proof in Appendix for the sake of completeness.

Let 𝔞 be any maximal abelian subspace in 𝔭. All such subspaces are mutually conjugate by an element of K_{o}. The dimension dim 𝔞 is called the real rank of G, denoted by ℝ-rank G. A subspace in 𝔤 conjugate to 𝔞 in G is called a maximally split abelian subspace. Let A be the connected group with Lie algebra 𝔞, M′ (resp. M) the normalizer (resp. centralizer) of 𝔞 in K. Let

𝔤(𝔞; α) =^{def} {X ∈ 𝔤; [H, X] = α(H)X for any H ∈ 𝔞}

for α ∈ 𝔞^{∗}, and

Σ ≡ Σ(𝔤, 𝔞) =^{def} {α ∈ 𝔞^{∗}; 𝔤(𝔞; α) ≠ 0} \ {0}

be the restricted root system for (𝔤, 𝔞). Let W ≡ W (𝔤; 𝔞) ≃ M′/M be the corresponding Weyl group of Σ (often called the little Weyl group). The simultaneous diagonalization of ad(𝔞)|_{𝔤} gives the decomposition

𝔤 = 𝔤(𝔞; 0) + ∑_{α∈Σ} 𝔤(𝔞; α),

𝔤(𝔞; 0) = 𝔪 + 𝔞.

Let p_{α} : 𝔤 → 𝔤(𝔞; α) be the corresponding projection for each element α ∈ Σ ∪ {0}. The subset 𝔞′ = {H ∈ 𝔞 : α(H) ≠ 0 for any α ∈ Σ} of regular elements consists of the complement of finitely many hyperplanes, and (the closures of) its components are called (closed) Weyl chambers.

For Y ∈ 𝔞, t ∈ ℝ, put

Ξ(Y; t) =^{def} {α ∈ Σ ∪ {0} : α(Y) ≥ t}.

It easy to see that Ξ(Y; t) = Ξ(Z; t) for all t ∈ ℝ if and only if Y = Z (∈ 𝔞) when the center of G is compact. Set

𝔭(Y) =^{def} ∑_{α∈Ξ(Y;0)} 𝔤(𝔞; α),

𝔩(Y) =^{def} Z_{𝔤}(Y) = ∑_{α(Y)=0} 𝔤(𝔞; α),

𝔫(Y) =^{def} ∑_{α(Y)>0} 𝔤(𝔞; α),

P (Y) =^{def} N_{G}(𝔭(Y)),
L(Y) =^{def} N_{P (Y)}(𝔩(Y)) = P (Y) ∩ θP (Y),
N(Y) =^{def} exp(𝔫(Y)).

Then 𝔭(Y) (resp. P (Y)) is a parabolic subalgebra (resp. parabolic subgroup) with Levi decomposition

(2.8)  𝔭(Y) = 𝔫(Y) + 𝔩(Y) (resp. P (Y) = N(Y) · L(Y) ).

Since G is contained in a connected complexification G_{ℂ},

(2.9)  L(Y) ≃ Z_{G}(Y).

Then the following lemma is well-known (cf. [War] Ch.1):

Lemma(2.10). Let Y, Z ∈ 𝔞 be contained in the same closed Weyl chamber for Σ(𝔤, 𝔞). If a ∈ G satisfies Ad(a) 𝔭(Y) = 𝔭(Z), then

𝔭(Y) = 𝔭(Z) and a ∈ P (Y) = P (Z).

Lemma(2.11). Let Y ∈ 𝔞 and a ∈ P (Y). Then

( ⨁_{β(Y)=t} p_{β}) (Ad(a) ∑_{α(Y)=t} 𝔤(𝔞; α)) = ∑_{α(Y)=t} 𝔤(𝔞; α),

( ⨁_{β(Y)=t} p_{β}) (Ad(a) ∑_{α(Y)>t} 𝔤(𝔞; α)) = 0.

Proof of Lemma(2.11). Let a = n · l (n ∈ N(Y), l ∈ L(Y)) be a Levi decomposition corresponding to (2.8). Since l centralizes Y by (2.9), we have

Ad(l) ∑_{α(Y)=t} 𝔤(𝔞; α) = ∑_{α(Y)=t} 𝔤(𝔞; α)

for any t ∈ ℝ. So we may and do assume a = n ∈ N(Y) to prove the lemma. If X_{α} ∈ 𝔤(𝔞; α) (α ∈ Σ ∪ {0}) and n ∈ N(Y), we have

Ad(n) X_{α} − X_{α} ∈ ∑_{β} 𝔤(𝔞; α + β),

where the sum is taken over the roots of 𝔫(Y) for 𝔞, namely {β ∈ Σ : β(Y) > 0}. Now the lemma follows. □

3. A Lemma (Abelian Case)

Throughout this section G is a real reductive linear group and we retain notations as in §2. We shall show a key lemma(3.1) to the criterion of the properness of the action of a reductive subgroup on a homogeneous space of reductive type (see Theorem(4.1)).

Lemma(3.1). Let 𝔞_{1}, 𝔞_{2} be two subspaces in 𝔞, and denote by A_{1}, A_{2} the analytic subgroups corresponding to 𝔞_{1}, 𝔞_{2} respectively. Then the following two conditions are equivalent: (3.1.1) For any compact subset S in G, SA_{1}S^{−1} ∩ A_{2} is compact. (3.1.2) For any w ∈ W (𝔤, 𝔞), w · 𝔞_{1} ∩ 𝔞_{2} = {0}.

Proof. We may and do assume that the center of G is compact. Since the Weyl group W (𝔤, 𝔞) ≃ N_{K}(𝔞)/Z_{K}(𝔞) = M′/M, it is easy to see that (3.1.2) follows from (3.1.1).

Conversely, assume SA_{1}S^{−1} ∩ A_{2} is not compact with some compact subset S in G. Then the following claim holds:

Claim(3.2). With the notation in Lemma(3.1), if there is a compact subset S in G such that SA_{1}S^{−1} ∩ A_{2} is not compact, then there are sequences

a_{n}, b_{n} ∈ G, Y_{n} ∈ 𝔞_{1}, Z_{n} ∈ 𝔞_{2}, t_{n} ∈ ℝ_{+}, (n ∈ ℕ)

and there are

a, b ∈ G, Y ∈ 𝔞_{1}, Z ∈ 𝔞_{2}, w ∈ W (𝔤.𝔞)

such that

lim_{n→∞} t_{n} = ∞

lim_{n→∞} a_{n} = a, lim_{n→∞} b_{n} = b,

lim_{n→∞} Y_{n} = Y, lim_{n→∞} Z_{n} = Z,

a_{n} = exp(t_{n}Z_{n}) b_{n} exp(−t_{n}w · Y_{n}),

where w · Y and Z are contained in the same closed Weyl chamber for Σ(𝔤, 𝔞).

The proof of this claim will be given soon. Let us continue the proof of the lemma. Using the above claim and its notations, we have

Ad(a_{n})X_{α} = Ad(exp(t_{n}Z))Ad(b_{n})Ad(exp(−t_{n} w · Y_{n}))X_{α}

= ∑_{β} exp t_{n}(β(Z) − α(w · Y_{n})) p_{β}(Ad(b_{n})X_{α}).

for any root vector X_{α} ∈ 𝔤(𝔞; α) with α ∈ Σ∪{0}. Since the set {Ad(b_{n})X_{α}; n ∈ ℕ} is bounded, we have

Ad(a)X_{α} ∈ ∑_{β∈Ξ(Z;α(w·Y))} 𝔤(𝔞; β).

Therefore for any t ∈ ℝ,

Ad(a) ∑_{α∈Ξ(w·Y;t)} 𝔤(𝔞; α) ⊂ ∑_{β∈Ξ(Z;t)} 𝔤(𝔞; β).

Similarly the equation

a_{n}^{−1} = exp(t_{n}w · Y_{n})b_{n}^{−1} exp(−t_{n}Z_{n})

leads to

Ad(a^{−1}) ∑_{β∈Ξ(Z;t)} 𝔤(𝔞; β) ⊂ ∑_{α∈Ξ(w·Y;t)} 𝔤(𝔞; α).

Thus for any t ∈ ℝ,

(3.3)  Ad(a) ∑_{α∈Ξ(w·Y;t)} 𝔤(𝔞; α) = ∑_{β∈Ξ(Z;t)} 𝔤(𝔞; β).

In particular putting t = 0 in (3.3), we have

Ad(a) 𝔭(w · Y) = 𝔭(Z).

As w · Y and Z are contained in the same closed Weyl chamber, we have

a ∈ P (w · Y) = P (Z),

from Lemma(2.10). Now operating the projection

⨁_{β(w·Y)=t} p_{β} : 𝔤 ⟶ ∑_{β(w·Y)=t} 𝔤(𝔞; β)

to the both sides of (3.3) for fixed t ∈ ℝ, we get

∑_{α(w·Y)=t} 𝔤(𝔞; α) = ∑_{β(Z)≥t=β(w·Y)} 𝔤(𝔞; β).

from Lemma(2.11). This equation implies

{α ∈ Σ ∪ {0} : α(w · Y) = t} ⊂ Ξ(Z; t).

In particular we have Ξ(w · Y; t) ⊂ Ξ(Z; t) for all t ∈ ℝ. The converse inclusion is obtained in the same way by operating ⨁_{β(Z)=t} p_{β} to the equation ∑_{α∈Ξ(w·Y;t)} 𝔤(𝔞; α) = Ad(a)^{−1} ∑_{β∈Ξ(Z;t)} 𝔤(𝔞; β). Hence we have proved

Ξ(w · Y; t) = Ξ(Z; t) for all t ∈ ℝ.

This relation implies Z = w · Y and so the condition (3.1.2) holds. The proof of Lemma(3.1) is now completed except for showing Claim(3.2). □

Proof of Claim(3.2). Replacing S by KSK if necessary, we may assume S is bi-K-invariant. From the assumption and the compactness of S, we can choose sequences

a_{n}, b_{n} ∈ S, Y_{n} ∈ 𝔞_{1}, Z_{n} ∈ 𝔞_{2}, t_{n} ∈ ℝ_{+}, u_{n} ∈ ℝ_{+}, (n ∈ ℕ)

such that

lim_{n→∞} u_{n} = ∞

lim_{n→∞} a_{n} = a, lim_{n→∞} b_{n} = b,

lim_{n→∞} Y_{n} = Y, lim_{n→∞} Z_{n} = Z,

a_{n} = exp(t_{n}Z_{n}) b_{n} exp(−u_{n}Y_{n}),

with some a, b ∈ G, Y ∈ 𝔞_{1} \ {0}, Z ∈ 𝔞_{2} \ {0}. Choose a positive system Σ^{+} of Σ making Z a dominant element. Take m_{w} ∈ N_{K}(𝔞) representing an element w ∈ W (𝔤; 𝔞) such that w·Y is dominant for Σ^{+}. Replacing a_{n}, b_{n} by a_{n}m_{w}^{−1}, b_{n}m_{w}^{−1} (with the same notations), we have

a_{n} = exp(t_{n}Z_{n})b_{n} exp(−u_{n} w · Y_{n}).

Therefore for each root vector X_{α} ∈ 𝔤(𝔞; α),

(3.4)  Ad(a_{n})X_{α} = ∑_{β∈Σ∪{0}} exp(t_{n}β(Z_{n}) − u_{n}α(Y_{n})) p_{β}(Ad(b_{n})X_{α}).

Let us show that \{\frac{t_{n(k)}}{u_{n(k)}} : n ∈ ℕ\} is bounded from 0 and ∞. In fact, suppose

there were subsequences n(k) (k ∈ ℕ) such that lim_{k→∞} \frac{t_{n(k)}}{u_{n(k)}} = ∞. Choose β ∈ Σ

such that β(Z) > 0 which exists because Z ≠ 0. Then the existence of the limit of (3.4) as n(k) → ∞ requires that p_{β}(Ad(b)X_{α}) = 0 for any α ∈ Σ ∪ {0}, which leads to a contradiction because Ad(b) is invertible. Similarly, suppose there were

subsequences n(k) (k ∈ ℕ) such that lim_{k→∞} \frac{t_{n(k)}}{u_{n(k)}} = 0. Choose α ∈ Σ such that

α(Y) > 0 which exists because Y ≠ 0. Then the existence of the limit of (3.4) as n(k) → ∞ requires that Ad(a)X_{α} = lim_{k→∞} Ad(a_{n(k)})X_{α} = 0, which leads to a

contradiction because Ad(a) is invertible. Thus we have shown that \{\frac{t_{n}}{u_{n}} : n ∈ ℕ\}

is bounded from 0 and ∞. Therefore by taking subsequences we may assume

lim_{n→∞} \frac{t_{n}}{u_{n}} = C with some positive constant C. Replacing u_{n}, Y_{n} and Y by t_{n}, \frac{u_{n}}{t_{n}} Y_{n}

and \frac{1}{C} Y respectively, we get the claim. □

4. Main Results

We consider the situation that a reductive subgroup H_{1} in G acts on a homogeneous space G/H_{2} of reductive type. First we give a criterion of the properness of this action.

Theorem(4.1). Let H_{1}, H_{2} be reductive subgroups in a real reductive linear group G (Definition (2.6)). Let 𝔞(H_{1}), 𝔞(H_{2}) and 𝔞 be maximally split abelian subspaces in 𝔥_{1}, 𝔥_{2} and 𝔤 respectively. Fix g_{i} ∈ G such that 𝔞_{i} =^{def} Ad(g_{i})𝔞(H_{i}) ⊂ 𝔞 (i = 1, 2). Then the following three conditions on {H_{1}, H_{2}} are equivalent: (4.1.1) H_{1} acts on G/H_{2} properly. (4.1.2) H_{2} acts on G/H_{1} properly. (4.1.3) For any w ∈ W (𝔤; 𝔞), w · 𝔞_{1} ∩ 𝔞_{2} = {0}.

Proof. First note that any of the condition (4.1.1)-(4.1.3) is independent of the choice of g_{i} and does not change by replacing H_{i} by g_{i}H_{i}g_{i}^{−1}. Therefore we can and do assume that g_{i} = e (i = 1, 2) and that there is a Cartan involution θ such that θH_{i} = H_{i} (i = 1, 2).

Since the role of H_{i} (i = 1, 2) are the same in (4.1.3), we only have to prove that (4.1.2) is equivalent to (4.1.3). From the definition of a proper action, (4.1.2) is equivalent to the compactness of the set

{h ∈ H_{2} : hSH_{1} ∩ SH_{1} ≠ ∅} = SH_{1}S^{−1} ∩ H_{2}

for every compact set S in G. In this condition we only have to treat the case S is bi-K-invariant. Put A_{i} := exp(𝔞_{i}). If S is bi-K-invariant, the Cartan decomposition of H_{i} ([He] Ch.IX Thm.1.1) gives

SH_{1}S^{−1} ∩ H_{2} = S(K ∩ H_{1})A_{1}(K ∩ H_{1})S^{−1} ∩ (K ∩ H_{2})A_{2}(K ∩ H_{2})
= (K ∩ H_{2})(SA_{1}S^{−1} ∩ A_{2})(K ∩ H_{2}).

Using the Cartan decomposition again, SH_{1}S^{−1} ∩ H_{2} is compact iff SA_{1}S^{−1} ∩ A_{2} is compact. Hence the proof of Theorem reduces to the preceding Lemma(3.2) in an abelian case. □

As an immediate corollary to Theorem(4.1), we have

Corollary(4.2). Let G′ be a reductive subgroup in G acting properly on a homogeneous space G/H of reductive type. Then

ℝ-rankG′ + ℝ-rankH ≤ ℝ-rankG.

This estimate is best possible. That is

Corollary(4.3). Let G/H be a homogeneous space of reductive type. Then there exists a subgroup G′ reductive in G such that G′ acts properly on G/H and that ℝ-rankG′ + ℝ-rankH = ℝ-rankG.

In fact Theorem(4.1) guarantees a trivial choice of G′: we can take G′ := exp 𝔟, where 𝔟 is any (ℝ-rank G − ℝ-rank H)-dimensional subspace complementary to W (𝔤; 𝔞) · 𝔞(H) in 𝔞.

It does depend on H and G whether a larger subgroup (with the same real rank) than the above G′ can act properly on G/H or not . Although it is interesting to classify the maximal ones among such subgroups based on Theorem(4.1), we shall not go into here. Anyway, now we can tell explicitly when Calabi-Markus phenomenon occurs in a homogeneous space of reductive type:

Corollary(4.4). Let G/H be a homogeneous space of reductive type. Then the following three conditions are equivalent: (4.4.1) A subgroup which can act properly discontinuously and freely on G/H must be finite. (4.4.1)′ Only finite subgroup can act properly discontinuously on G/H. (4.4.2) ℝ-rank G = ℝ-rank H.

Proof. If k := ℝ-rank G − ℝ-rank H > 0, then there is a subgroup Γ isomorphic to ℤ^{k} consisting of semisimple elements in G such that Γ acts properly discontinuously and freely on G/H by Corollary(4.3). Thus (4.4.1) → (4.4.2) is proved. (4.4.1)′ → (4.4.1) is trivial. Although (4.4.2) → (4.4.1)′ is immediately deduced from a known sufficient condition for Calabi-Markus phenomena ([Ku] Theorem A.1.2, see also [Wo]), we review it because it is elementary but instructive: Suppose ℝrankG = ℝ-rankH. Then a maximally split abelian subspace 𝔞(H) in 𝔥 is also a maximal one in 𝔤, and thus G = K exp 𝔞(H)K = KHK, which implies that {g ∈ G; g(K/H ∩ K) ∩ (K/H ∩ K) ≠ ϕ in G/H} = G. Since K/H ∩ K is compact, this implies (4.4.1)′. □

Example(4.5). Only finite subgroup in GL(n, ℂ) can act properly discontinuously on GL(n, ℂ)/GL(n, ℝ).

Remark(4.6).

1) Kulkarni proved the necessity of (4.4.2) in a special case: that is, when G/H = SO(p + 1, q)/SO(p, q), (4.4.1) holds if p < q. One should note that p < q is equivalent to ℝ-rankG > ℝ-rankH in this case. But our proof is different from his method ([Ku]).

2) The above corollary does not exclude the existence of an infinite abstract group acting properly discontinuously on G/H. For instance G/H = SL(2, ℝ)/A, (A is the diagonal matrix group) is diffeomorphic to S^{1} × ℝ, and so admitting a properly discontinuous and free action of ℤ.

3) When G/H is a semisimple symmetric space, the real rank condition in Corollary(4.4) is equivalent to the condition that the ‘dual symmetric space’ G^{d}/H^{d} has a nonempty discrete series (ref. [FJ]).

Following the approach stated in Introduction, we want a reductive subgroup G′ as large as possible acting on G/H. The extreme case − where G′\G/H is compact − is characterized by a simple condition:

Theorem(4.7). Let H_{i} (i = 1, 2) be reductive subgroups in a real reductive linear group G (Definition(2.6)). Under the equivalent conditions (4.1.1)-(4.1.3), the following four conditions are equivalent: (4.7.1) H_{1}\G/H_{2} is compact. (4.7.2) There exists a discrete group Γ_{1} in H_{1} so that Γ_{1}\G/H_{2} is a compact smooth manifold. (4.7.2)′ There exists a discrete group Γ_{2} in H_{2} so that H_{1}\G/Γ_{2} is a compact smooth manifold. (4.7.3) d(G) = d(H_{1}) + d(H_{2}).

The proof of Theorem(4.7) together with Proposition(4.10) below will be given at the end of §5. Theorem(4.1) and Theorem(4.7) give a method to get examples of a (not necessarily Riemannian) homogeneous space admitting a uniform lattice: namely, find the triplet {G, H_{1}, H_{2}} which satisfies the criteria (4.1.3) and (4.7.3), and then there exists a uniform lattice in G/H_{i} for i = 1, 2. Notice that if H_{2} is compact, we can always choose H_{1} = G. The next one is also rather stupid.

Example(4.8). Let G′ be a real reductive linear group. Then G/H_{2} := \underbrace{G′ × G′ × ⋯ × G′}_{n−times} /ΔG′,

where ΔG′  :=  {(g, …, g)  ∈  G  :  g  ∈  G′},

admits a uniform lattice because we can choose H_{1} := G′ × G′ × ⋯ × G′ × {e}.

The following examples are remarkable:

Proposition(4.9). Either of the following triplets {G, H_{1}, H_{2}} satisfies both (4.1.1)- (4.1.3) and (4.7.1)-(4.7.3). Therefore G/H_{i} (i = 1, 2) admits a uniform lattice as well as a non-uniform lattice.

1) G = U(2, 2n), H_{1} = Sp(1, n), H_{2} = U(1) × U(1, n),

2) G = SO(2, 2n), H_{1} = U(1, n), H_{2} = SO(1, 2n),

3) G = SO(4, 4n), H_{1} = Sp(1, n), H_{2} = SO(3, 4n).

This Proposition can be easily checked by the criteria (4.1.3) and (4.7.3). Among these examples, G/H_{2} in 2) and 3) were previously obtained in Theorem 6.1 in [Ku].

Finally we give another evidence (cf. Corollary(4.4)) that not so many homogeneous spaces of reductive type have uniform lattices:

Proposition(4.10). Let H be a maximal rank reductive subgroup in a real reductive linear group G. Then G/H admits a uniform lattice only if

rankK = rankH ∩ K.

Recall that the rank of a reductive group G, denoted by rank G, is the dimension of a Cartan subalgebra of 𝔤 ⊗ ℂ over ℂ. The above result is somewhat stronger than the one which K.Ono and the author have recently obtained (see Corollary 5 in [K-O]).

Example(4.11). Sp(2n, ℝ)/Sp(n, ℂ) has no uniform lattice but admits a properly discontinuous and free action of a subgroup isomorphic to ℤ^{n}.

5. Application of Cohomological Dimension Theory

First we review some notations concerning the cohomology of an abstract group. General references are [Bi], [C-E] and [Ser].

Let Γ be an abstract group, R a commutative ring with 1 ≠ 0, R[Γ] the corresponding group ring. For each left R[Γ]-module A, the cohomology groups of Γ with coefficients in A are defined by

(5.1)  H^{q}(Γ, A) =^{def} Ext_{ℤ[Γ]}^{q}(ℤ, A) ≃ Ext_{R[Γ]}^{q}(R, A),

where R (resp. ℤ) is regarded as a left R[Γ] (resp. ℤ[Γ]) module with trivial action of Γ. For the second isomorphism in (5.1), see [C-E] Ch.X §3.4. The cohomological dimension of Γ over R denoted by cd_{R}(Γ) is the projective dimension of R as a left R[Γ]-module. Equivalently,

cd_{R}(Γ) = sup{n ∈ ℕ : H^{n}(Γ; A) ≠ 0 for some left R[Γ]−module A}.

Following Serre (see [Ser]), we call Γ virtually torsionless iff Γ has a torsionless subgroup of finite index. Then the following result is due to Selberg ([Sel] Lemma 8):

Lemma(5.2). A finitely generated matrix group is virtually torsionless.

From now on, we shall restrict ourselves to the case when R = ℝ, the field of real numbers. This suffices for our application in this paper.

In this case, cd_{ℝ}Γ′ = cd_{ℝ}Γ for any subgroup Γ′ of finite index in Γ (see [Ser] Thm.1, [Bi] §5.4).

Lemma(5.3). Let G/H be a homogeneous space of reductive type (Definition(2.6)). Let Γ be a virtually torsionless discrete subgroup of G. Set

S := dim(K/H ∩ K), N := cd_{ℝ}Γ.

1) (Serre) N < ∞

2) If Γ acts properly discontinuously on G/H, then there is a subgroup Γ′ of finite index in Γ such that Γ′ acts properly discontinuously and freely on G/H and that there is a natural isomorphism

(5.3.1)  H^{N}(Γ′; A) ≃ H^{N+S}(Γ′\G_{o}/H_{o}; A)

for any left ℝ[Γ′]-module A. Here A is regarded as a local coefficient system on Γ′\G_{o}/H_{o} in the right hand.

Proof. 1) is proved in [Ser]. In fact the spectral sequence (5.4) below collapses when H = K.

Suppose Γ act properly discontinuously on G/H. Take any torsion free subgroup Γ′ of finite index in Γ ∩ G_{o}. Then Γ′ also acts freely on G/H (and G_{o}/H_{o}) because the action is properly discontinuous. There is a well-known first quadrant spectral sequence corresponding to the covering G_{o}/H_{o} → Γ′\G_{o}/H_{o} ([C-E] Ch.XVI §9)

(5.4)  E_{n}^{p,q} ⟹ H^{p+q}(Γ′\G_{o}/H_{o}; A),

with E_{2} term

E_{2}^{p,q} ≃ H^{p}(Γ′; H^{q}(G_{o}/H_{o}; A))

The differential d_{n} has bidegree (n, 1 − n). Here the action of Γ′ on H^{q}(G_{o}/H_{o}; A) is the diagonal one induced from the action of Γ × Γ on H^{q}(G_{o}/H_{o}; A). Since Γ′ is contained in G_{o}, this action agrees the action on the second factor A alone. As Lemma(2.7) assures that G_{o}/H_{o} has the same homotopy type with K_{o}/H_{o} ∩ K_{o}, we have the following Γ′-module isomorphism:

H^{q}(G_{o}/H_{o}; A) ≃ H^{q}(K_{o}/H_{o} ∩ K_{o}; A) ≃ \{ \begin{matrix}0, & if q > S \\ A, & if q = S. \end{matrix}

Then the spectral sequence (5.4) and the definition of cd_{ℝ}Γ′(= cd_{ℝ}Γ = N) yield (5.3.1). □

Now the same argument in [Ser] Prop.18 (cf. [Ku] Thm.2.1) leads to a cohomological restriction on Γ from the topology of a smooth manifold Γ′\G_{o}/H_{o} by (5.3.1). Recall that we have defined d(G) by the dimension of the Riemannian symmetric space associated to G ((2.5)).

Corollary(5.5). Let Γ be a discrete subgroup in G acting properly discontinuously on a homogeneous space G/H of reductive type.

1) If Γ\G/H is compact, then

a) Γ is virtually torsionless,

b) cd_{ℝ}Γ = d(G) − d(H).

Fix a torsion free subgroup Γ′ of finite index in Γ ∩ G_{o}.

c) dim_{ℝ} H^{j}(Γ′; ℝ) < ∞ for all j ∈ ℤ,

d) χ(Γ′) χ(K_{o}/H_{o} ∩ K_{o}) = χ(Γ′\G/H). Here χ(Γ′) =^{def} ∑_{j=0}^{N}(−1)^{j} dim_{ℝ} H^{j}(Γ′; ℝ), χ(M) denotes the Euler number of a compact orientable manifold M.

2) If Γ\G/H is noncompact and if Γ is virtually torsionless, then cd_{ℝ}Γ ≤ d(G) − d(H) − 1.

Proof. a) is a direct consequence of Lemma(2.1) and Lemma(5.2). b) and 2) is deduced from (5.3.1) and from the following well-known Lemma(5.6) (This is proved by the Poincaré duality). Notice that S = dim(G/H)−(d(G)−d(H)) in the notation of Lemma(5.3). c) is shown by induction from N to 0 by using the spectral sequence (5.4), and d) is the Euler-Poincaré principle. □

Lemma(5.6). For any local system S on a noncompact manifold M, H^{j}(M, S) = 0 if j ≥ dim M.

Now we are ready to prove Proposition(4.10).

Proof of Proposition(4.10). The proof is essentially the same as in Corollary 5 in [K-O]. What we must show is only the fact that G/H is a θ-stable homogeneous space in the sense of [K-O], Definition(3.3). This follows from the fact that an analytic subgroup H_{ℂ} in G_{ℂ} with Lie algebra 𝔥 ⊗ ℂ is closed because of the rank condition rankH = rankG. □

Finally let us prove Theorem(4.7) as an application of Corollary(5.5).

Proof of Theorem(4.7). Since a real reductive linear group has a uniform lattice ([Bo]), (4.7.1), (4.7.2) and (4.7.2)′ are equivalent from Lemma(2.3). Fix any torsion free discrete subgroup Γ_{1} in H_{1}. Apply Corollary(5.5) with G = H_{1} and H = {e}, we have cd_{ℝ}Γ_{1} ≤ d(H_{1}) and the equality holds iff Γ_{1}\H_{1} is compact. Apply Corollary(5.5) again with H = H_{2}, we have cd_{ℝ}Γ_{1} ≤ d(G)−d(H_{2}) and the equality holds iff Γ_{1}\G/H_{2} is compact. If (4.7.2) holds, then d(G) = cd_{ℝ}Γ_{1} + d(H_{2}) ≤ d(H_{1}) + d(H_{2}). On the other hand, the assumption (4.1.3) implies that d(G) ≥ d(H_{1}) + d(H_{2}), showing (4.7.3). Conversely if (4.7.3) holds, any uniform lattice Γ_{1} in H_{1} is cocompact in G/H_{2} because cd_{ℝ}Γ_{1} = d(H_{1}) = d(G) − d(H_{2}). Now the Theorem follows. □

6. Appendix

Let H be a closed subgroup reductive in a real reductive linear group G (Definition (2.6)). Fix a Cartan involution θ of G such that H = (H ∩ K) exp(𝔥 ∩ 𝔭).

Lemma(6.1). Retain notations as above. Then the mapping

π : (𝔥 ∩ 𝔭) + (𝔮 ∩ 𝔭) ∋ (X, Y) ↦ exp X exp Y · o ∈ G/K

gives a surjective diffeomorphism.

Proof. The identity mapping of H into G induces an inclusion H/H ∩ K ↪ G/K. Since H is a closed subgroup in G, H/H ∩ K is also a closed submanifold in G/K. Identifying as usual the tangent space T (G/K)_{o} with 𝔭, we have T (H/H∩K)_{o} ≃ 𝔥∩𝔭. As the geodesics through o have the form exp(tX)·o (t ∈ ℝ) where X is a general vector in 𝔭, this is tangent to H/H ∩ K at o if and only if X ∈ 𝔥∩𝔭. Since H acts on G/K isometrically, it follows that H/H∩K = exp(𝔥∩𝔭)·o is totally geodesic in G/K. Fix an element X of 𝔥∩𝔭, and put p = exp(X)·o ∈ G/K and S^{⊥}(p) =^{def} exp(X) exp(𝔮 ∩ 𝔭) · o ⊂ G/K. Pull back the tangent space at p by L_{exp(X)∗}^{−1} : T (G/K)_{p} → 𝔭, and we have

L_{exp(X)∗}^{−1}(T (H/H ∩ K)_{p}) = A_{X}(𝔥 ∩ 𝔭),

L_{exp(X)∗}^{−1}(T (S^{⊥}(p))_{p}) = 𝔮 ∩ 𝔭,

where A_{X} =^{def} ∑^{∞} ; n=0 \frac{(adX)^{2n}}{(2n + 1)!}_{|𝔭} ∈ GL(𝔭) ([He] Ch.1 Theorem 13.3 and Ch.4 Theorem

4.1). Since X ∈ 𝔥 ∩ 𝔭, ad(X)^{2} preserves 𝔥 ∩ 𝔭 and therefore A_{X}(𝔥 ∩ 𝔭) = 𝔥 ∩ 𝔭. As 𝔥 ∩ 𝔭 and 𝔮 ∩ 𝔭 are orthogonal to each other with respect to ⟨ , ⟩, S^{⊥}(p) is a submanifold made of the geodesics in G/K which are perpendicular to H/H ∩ K at p. Thus we have (see [He] Ch.1 Theorem 14.6)

G/K = ∐_{X∈𝔥∩𝔭} S^{⊥}(p)

= ∐_{X∈𝔥∩𝔭} exp(X) exp(𝔮 ∩ 𝔭) · o.

Hence π is bijective. □

Now let us prove Lemma(2.7).

Proof of Lemma(2.7). From Lemma(6.1), we have

G = K exp(𝔮 ∩ 𝔭) exp(𝔥 ∩ 𝔭)
= K exp(𝔮 ∩ 𝔭) exp(𝔥 ∩ 𝔭) (H ∩ K)
= K exp(𝔮 ∩ 𝔭) H.

Let f(k_{1}, Z_{1}) = f(k_{2}, Z_{2}) for k_{1}, k_{2} ∈ K and Z_{1}, Z_{2} ∈ 𝔮 ∩ 𝔭. Then there are X ∈ 𝔥 ∩ 𝔭 and h ∈ H ∩ K such that k_{1} exp(Z_{1}) exp(X)h = k_{2} exp(Z_{2}). Therefore k_{1}h exp(Ad(h)^{−1}(Z_{1})) exp(Ad(h)^{−1}(X)) = k_{2} exp(Z_{2}). Since Ad(h)^{−1}(Z_{1}) ∈ 𝔮 ∩ 𝔭 and Ad(h)^{−1}(X) ∈ 𝔥 ∩ 𝔭, we have k_{1}h = k_{2}, Ad(h)^{−1}(Z_{1}) = Z_{2} and X = 0, from the uniqueness of the decomposition G = K exp(𝔮 ∩ 𝔭) exp(𝔥 ∩ 𝔭). □

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