Isometries, rigidity and universal covers

Benson Farb and Shmuel Weinberger ^{∗}

December 31, 2006

1 Introduction

The goal of this paper is to describe all closed, aspherical Riemannian manifolds M whose universal covers M̃ have have a nontrivial amount of symmetry. By this we mean that Isom(M̃) is not discrete. By the well-known theorem of Myers-Steenrod [MS], this condition is equivalent to [Isom(M̃) : π_{1}(M)] = ∞. Also note that if any cover of M has a nondiscrete isometry group, then so does its universal cover M̃.

Our description of such M is given in Theorem 1.2 below. The proof of this theorem uses methods from Lie theory, harmonic maps, large-scale geometry, and the homological theory of transformation groups.

The condition that M̃ have nondiscrete isometry group appears in a wide variety of problems in geometry. Since Theorem 1.2 provides a taxonomy of such M, it can be used to reduce many general problems to verifications of specific examples. Actually, it is not always Theorem 1.2 which is applied directly, but the main subresults from its proof. After explaining in §1.1 the statement of Theorem 1.2, we give in §1.2 a number of such applications. These range from new characterizations of locally symmetric manifolds, to the classification of contractible manifolds covering both compact and finite volume manifolds, to a new proof of the Nadel-Frankel Theorem in complex geometry.

1.1 Statement of the general theorem

The basic examples of closed, aspherical, Riemannian manifolds whose universal covers have nondiscrete isometry groups are the locally homogeneous (Riemannian) manifolds M, i.e. those M whose universal cover admits a transitive Lie group action whose isotropy subgroups are maximal compact. Of course one might also take a product of such a manifold with an arbitrary manifold. To find nonhomogeneous examples which are not products, one can do the following construction.

∗Both authors are supported in part by the NSF.

Example 1.1. Let F → M → B be any Riemannian fiber bundle with the induced path metric on F locally homogeneous. Let f : B → R^{+} be any smooth function. Now at each point of M lying over b, rescale the metric in the tangent space T M_{b} = T F_{b} ⊕ T B_{b} by rescaling T F_{b} by f(b). Almost any f gives a metric on M with dim(Isom(M̃)) > 0 but with M̃ not homogeneous, indeed with each Isom(M̃)-orbit a fiber. This construction can be further extended by scaling fibers using any smooth map from B to the moduli space of locally homogeneous metrics on F; this moduli space is large for example when F is an n-dimensional torus.

Hence we see that there are many closed, aspherical, Riemannian manifolds whose universal covers admit a nontransitive action of a positive-dimensional Lie group. The following general result says that the examples described above exhaust all the possibilities for such manifolds.

Before stating the general result, we need some terminology. A Riemannian orbifold B is a smooth orbifold where the local charts are modelled on quotients V/G, where G is a finite group and V is a linear G-representation endowed with some G-invariant Riemannian metric. The orbifold B is good if it is the quotient of V by a properly discontinuous group action.

A Riemannian orbibundle is a smooth map M ⟶ B from a Riemannian manifold to a Riemannian orbifold locally modelled on the quotient map p : V ×_{G} F ⟶ V/G, where F is a fixed smooth manifold with smooth G-action, and where V × F has a G-invariant Riemannian metric such that projection to V is an orthogonal projection on each tangent space. Note that in this definition, the induced metric on the fibers of a Riemannian orbibundle may vary, and so a Riemannian orbibundle is not a fiber bundle structure in the Riemannian category.

Theorem 1.2. Let M be a closed, aspherical Riemannian manifold. Then either Isom(M̃) is discrete, or M is isometric to an orbibundle

F ⟶ M ⟶ B  (1.1)

where:

• B is a good Riemannian orbifold.

• Each fiber F, endowed with the induced metric, is isometric to a closed, aspherical, locally homogeneous Riemannian n-manifold, n > 0 ^{1}.

1Recall that a manifold F is locally homogeneous if its universal cover is isometric to G/K, where G is a Lie group, K is a maximal compact subgroup, and G/K is endowed with a left G-invariant, K bi-invariant metric.

Note that B is allowed to be a single point.

One might hope that the Riemannian orbifold B in the conclusion of Theorem 1.2 could be taken to be a Riemannian manifold, at least after passing to a finite cover of M. This is not the case, however. In §6 we construct a Riemmanian manifold M with the property that M is a Riemannian orbibundle, fibering over a singular orbifold, but such that no finite cover of M fibers over a manifold; further, Isom(M̃) is not discrete. This seems to be the first known example of an aspherical manifold with a singular fibration that remains singular in every finite cover. In constructing M we produce a group Γ acting properly discontinuously and cocompactly by diffeomorphisms on R^{n}, but which is not virtually torsion-free.

1.2 Applications

We now explain how to apply Theorem 1.2 and its proof to a variety of problems in geometry. The proofs of these results will be given in §4 below.

Characterizations of locally symmetric manifolds. We begin with a characterization of locally symmetric manifolds among all closed Riemannian manifolds. The theme is that such manifolds are characterized by some simple properties of their fundamental group, together with the property that their universal covers have nontrivial symmetry (i.e. have nondiscerete isometry group). We say that a smooth manifold M is smoothly irreducible if M is not smoothly covered by a nontrivial finite product of smooth manifolds.

Theorem 1.3. Let M be any closed Riemannian n-manifold, n > 1. Then the following are equivalent:

1. M is aspherical, smoothly irreducible, π_{1}(M) has no nontrivial, normal abelian subgroup, and Isom(M̃) is not discrete.

2. M is isometric to an irreducible, locally-symmetric Riemannian manifold of nonpositive sectional curvature.

The idea here is to apply Theorem 1.2, or more precisely the main results in its proof, and then to show that if the base B were positive dimensional, the manifold M would not be smoothly irreducible; see §4.1 below.

Remark. The proof of Theorem 1.3 gives more: the condition that M is smoothly irreducible can be replaced by the weaker condition that M is not Riemannian covered by a nontrivial Riemannian warped product; see §4.1.

When M has nonpositive curvature, the Cartan-Hadamard Theorem gives that M is aspherical. For nonpositively curved metrics on M, Theorem 1.3 was proved by Eberlein in [Eb1, Eb2] ^{2}. While the differential geometry and dynamics related to nonpositive curvature are central to Eberlein’s work, for the most part they do not, by necessity, play a role in this paper.

Recall that the Mostow Rigidity Theorem states that a closed, aspherical manifold of dimension at least three admits at most one irreducible, nonpositively curved, locally symmetric metric up to homotheties of its local direct factors. For such locally symmetric manifolds M, Theorem 1.3 has the following immediate consequence:

Up to homotheties of its local direct factors, the locally symmetric metric on M is the unique Riemannian metric with Isom(M̃) not discrete.

Uniqueness within the set of nonpositively curved Riemannian metrics on M follows from [Eb1, Eb2]. This statement also generalizes the characterization in [FW] of the locally symmetric metric on an arithmetic manifold.

Combined with basic facts about word-hyperbolic groups, Theorem 1.3 provides the following characterization of closed, negatively curved, locally symmetric manifolds.

Corollary 1.4. Let M be any closed Riemannian n-manifold, n > 1. Then the following are equivalent:

1. M is aspherical, π_{1}(M) is word-hyperbolic, and Isom(M̃) is not discrete.

2. M is isometric to a negatively curved, locally symmetric Riemannian manifold.

Theorem 1.3 can also be combined with Margulis’s Normal Subgroup Theorem to give a simple characterization in the higher rank case. We say that a group Γ is almost simple if every normal subgroup of Γ is finite or has finite index in Γ.

Corollary 1.5. Let M be any closed Riemannian manifold. Then the following are equivalent:

1. M is aspherical, π_{1}(M) is almost simple, and Isom(M̃) is not discrete.

2. M is isometric to a nonpositively curved, irreducible, locally symmetric Riemannian manifold of (real) rank at least 2.

The above results distinguish, by a few simple properties, the locally symmetric manifolds among all Riemannian manifolds. We conjecture that a stronger, more quantitative result holds, whereby there is a kind of universal (depending only on π_{1}) constraint on the amount of symmetry of any Riemannian manifold which is not an orbibundle with locally symmetric fiber.

2Eberlein’s results are proved not just for lattices but more generally for groups satisfying the so-called duality condition (see [Eb1, Eb2]), a condition on the limit set of the group acting on the visual boundary.

Conjecture 1.6. The hypothesis “Isom(M̃) is not discrete” in Theorem 1.3, Corollary 1.5, and Corollary 1.4 can be replaced by: [Isom(M̃) : π_{1}(M)] > C, where C depends only on π_{1}(M).

We do not know how to prove Conjecture 1.6. However, we can prove it in the special case of a fixed manifold admitting a locally symmetric metric.

Theorem 1.7. Let (M, g_{0}) be a closed, irreducible, nonpositively curved locally symmetric n-manifold, n > 1. Then there exists a constant C, depending only on π_{1}(M), such that for any Riemannian metric h on M:

[Isom(M̃) : π_{1}(M)] > C if and only if h ∼ g_{0}

where ∼ denotes “up to homothety of direct factors”.

Manifolds with both closed and finite volume quotients. We can also apply our methods to answer the following fundamental question in Riemannian geometry: which contractible Riemannian manifolds X cover both a closed manifold and a (noncompact, complete) finite volume manifold?

This question has been answered for many (but not all) contractible homogeneous spaces X. Recall that a contractible (Riemannian) homogeneous space X is the quotient of a connected Lie group H by a maximal compact subgroup, endowed with a left-invariant metric. Mostow proved that solvable H admit only cocompact lattices, while Borel proved that noncompact, semisimple H have both cocompact and noncocompact lattices (see [Ra], Thms. 3.1, 14.1). The case of arbitrary homogeneous spaces is more subtle, and as far as we can tell remains open.

The following theorem extends these results to all contractible manifolds X. It basically states that if X covers both a compact and a noncompact, finite volume manifold, then the reason is that X is “essentially” a product, with one factor a homogeneous space which itself covers both types of manifolds.

To state this precisely, we define a warped Riemannian product to be a smooth manifold X = Y × Z where Z is a (locally) homogeneous space, f : Y ⟶ H(Z) is a smooth function with target the space H(Z) of all (locally) homogeneous metrics on Z, and the metric on X is given by

g_{X}(y, z) = g_{Y} ⊕ f(y)g_{Z}

We can now state the following.

Theorem 1.8. Let X be a contractible Riemannian manifold. Suppose that X Riemannian covers both a closed manifold and a noncompact, finite volume, complete manifold. Then X is isometric to a warped product Y × X_{0}, where Y is a contractible manifold (possibly a point) and X_{0} is a homogeneous space which admits both cocompact and noncocompact lattices. In particular, if X is not a Riemannian warped product then it is homogeneous.

Note that the factor Y is necessary, as one can see by taking the product of a homogeneous space with the universal cover of any compact manifold. We begin the deduction of Theorem 1.8 from the other results in this paper by noting that its hypotheses imply that Isom(Z) is nondiscrete, so that our general result can be applied.

Irreducible lattices in products. Let X = Y × Z be a Riemannian product. Except in obvious cases, Isom(Y) × Isom(Z) ↪ Isom(X) is a finite index inclusion. Recall that a lattice Γ in Isom(X) is irreducible if it is not virtually a product. Understanding which Lie groups admit irreducible lattices is a classical problem; see, e.g., [Ma], §IX.7. Eberlein determined in [Eb1, Eb2] the nonpositively curved X which admit irreducible lattices; they are essentially the symmetric spaces. The following extends this result to all contractible manifolds; it also provides another proof of Eberlein’s result.

Theorem 1.9. Let X be a nontrivial Riemannian product, and suppose that Isom(X) admits an irreducible, cocompact lattice. Then X is isometric to a warped Riemannian product X = Y × X_{0}, where Y is a contractible manifold (possibly a point), X_{0} is a positive dimensional homogeneous space, and X_{0} admits an irreducible, cocompact lattice.

As with Theorem 1.8, Theorem 1.9 is deduced from the other results in this paper by noting that its hypotheses imply that Isom(Z) is nondiscrete; see §4.6.

Compact complex manifolds. Our results on isometries also have implications for complex manifolds. Kazhdan conjectured that any irreducible bounded domain Ω which admits both a compact quotient M and a one-parameter group of holomorphic automorphisms must be biholomorphic to a bounded symmetric domain. Frankel [Fr1] first proved this for convex domains Ω, and subsequent work by Nadel [Na] and Frankel [Fr2], which we now recall, proved it in general.

The Bergman volume form on a bounded domain produces a metric on the canonical bundle so that the first Chern class satisfies c_{1}(M) < 0; equivalently, the canonical line bundle is ample. Hence Kazhdan’s conjecture is implied by (and, indeed, inspired) the following.

Theorem 1.10 (Nadel, Frankel). Let M be a compact, aspherical complex manifold with c_{1}(M) < 0. Then there is a holomorphic splitting M′ = M_{1} × M_{2} of a finite cover M′ of M, where M_{1} is locally symmetric and M_{2} is locally rigid (i.e. the biholomorphic automorphism group of the universal cover M̃_{2} is discrete.)

Theorem 1.10 was first proved in (complex) dimension two by Nadel [Na] and in all dimensions by Frankel [Fr2]. They do not require the asphericity of M, although this is of course the case for quotients of bounded domains. Complex geometry is an essential ingredient in their work.

In §4.8 we give a different proof of Theorem 1.10, using a key proposition from the earlier paper of Nadel [Na] together with (the proof of) our Theorem 1.2 below. In complex dimension two, we give a proof independent of both [Na] and [Fr2]. We do not see, however, how to use our methods without the asphericity assumption.

As with [Na] and [Fr2], our starting point is a theorem of Aubin-Yau, which gives that the biholomorphism group Aut(M̃) acts isometrically on a Kahler-Einstein metric lifted from M. Our proof shows that, at least in complex dimension two, this is the only ingredient from complex geometry needed to prove Kazhdan’s conjecture.

Remark. Nadel pointed out explicitly in Proposition 0.1 of [Na] that his methods would extend to prove Theorem 1.10 in all dimensions if one could prove that each isotropy subgroup of Aut(M̃)^{∘} were a maximal compact subgroup. The solution to this problem in the aspherical case is given in Claim IV of Section 2 below; it also applies outside of the holomorphic context as well.

Some additional applications. A number of the results from this paper generalize from closed, aspherical Riemannian manifolds to all closed Riemannian manifolds. In §5 we provide an illustrative example, Theorem 5.1, which seems to be the first geometric rigidity theorem for non-aspherical manifolds with infinite fundamental group.

In §4.7 below we give an application of our methods to the Hopf Conjecture about Euler characetristics of aspherical manifolds.

Finally, we would like to mention the work of K. Melnick in [Me], where some of the results here are extended from the Riemannian to the pseudo-Riemannian (especially the Lorentz) case. Melnick combines the ideas here with Gromov’s theory of rigid geometric structures, as well as methods from Lorentz dynamics.

Acknowledgements. A first version of the main results of this paper were proved in the Fall of 2002. We would like to thank the audiences of the many talks we have given since that time on the work presented here; they provided numerous useful comments. We are particularly grateful to the students in “Geometric Literacy” at the University of Chicago, especially to Karin Melnick for her corrections on an earlier version of this paper. We would like to thank Ralf Spatzier who, after hearing a talk on some of our initial results (later presented in [FW]), pointed out a connection with Eberlein’s work; this in turn lead us to the idea that a much more general result might hold. Finally, we would like to thank the excellent referees, whose extensive comments and suggestions greatly improved the paper.

2 Finding the orbibundle (proof of Theorem 1.2)

Our goal in this section is to prove Theorem 1.2. The starting point is the following wellknown classical theorem.

Theorem 2.1 (Myers-Steenrod, [MS]). Let M be a Riemannian manifold. Then Isom(M) is a Lie group, and acts properly on M. If M is compact then Isom(M) is compact.

Note that the Lie group Isom(M) in Theorem 2.1 may have infinitely many components; for example, let M be the universal cover of a bumpy metric on the torus.

Throughout this paper we will use the following notation:

M= a closed, aspherical Riemannian manifold Γ = π_{1}(M) X = M̃ = the universal cover of M I = Isom(X) = the group of isometries of X I_{0} = the connected component of I containing the identity Γ_{0} = Γ ∩ I_{0}

Here X is endowed with the unique Riemannian metric for which the covering map X → M is a Riemannian covering. Hence Γ acts on X isometrically by deck transformations, giving a natural inclusion Γ → I, where I = Isom(X) is the isometry group of X.

By Theorem 2.1, I is a Lie group, possibly with infinitely many components. Let I_{0} denote the connected component of the identity of I; note that I_{0} is normal in I. If I is discrete, then we are done, so suppose that I is not discrete. Theorem 2.1 then gives that the dimension of I is positive, and so I_{0} is a connected, positive-dimensional Lie group.

We have the following exact sequences:

1 ⟶ I_{0} ⟶ I ⟶ I/I_{0} ⟶ 1  (2.1)

and

1 ⟶ Γ_{0} ⟶ Γ ⟶ Γ/Γ_{0} ⟶ 1  (2.2)

We now proceed in a series of steps. Our first step is to construct what will end up as the locally homogeneous fibers of the orbibundle (1.1).

Claim I: The quotient I_{0}/Γ_{0} is compact.

Proof. Let Fr(X) denote the frame bundle over X. The isometry group I acts freely on Fr(X). The I_{0} orbits in Fr(X) give a smooth foliation of Fr(X) whose leaves are diffeomorphic to I_{0}. This foliation descends via the natural projection Fr(X) ⟶ Fr(M) to give a smooth foliation F on Fr(M), each of whose leaves is diffeomorphic to I_{0}/Γ_{0}. Thus we must prove that each of these leaves is compact.

The quotient of Fr(X) by the smallest subgroup of I containing both Γ and I_{0} is homeomorphic to the space of leaves of F. We claim that this quotient is a finite cover of Fr(X)/I.

To prove this, it is clearly enough to show that the natural injection Γ/Γ_{0} ⟶ I/I_{0} has finite index image.

To this end, we first recall the following basic principle of Milnor-Svarc (see, e.g., [H]). Let G be a compactly generated topological group, generated by a compact subspace S ⊂ G. Endow G with the word metric, i.e. let d_{G}(g, h) be defined to be the minimal number of elements of S needed to represent gh^{−1}; this is a left-invariant metric on G. Now suppose that G acts properly and cocompactly by isometries on a proper, geodesic metric space X. Then G is quasi-isometric to X, i.e. for any fixed basepoint x_{0} ∈ X, the orbit map G ⟶ X sending g to g · x_{0} satisfies the following two conditions:

• (Coarse Lipschitz): For some K, C > 0,

\frac{1}{K} d_{G}(g, h) − C ≤ d_{X}(g · x_{0}, h · x_{0}) ≤ Kd_{G}(g, h) + C

• (C-density) Nbhd_{C}(G · x) = X

While the standard proofs of this fact (see, e.g., [H]) usually assume that S is finite, they apply verbatim to the more general case of S compact.

Applying this principle, the cocompactness of the actions of both Γ and of I on X give that the inclusion Γ ⟶ I is a quasi-isometry. The quotient map I ⟶ I/I_{0} is clearly distance nonincreasing, and so the image Γ/Γ_{0} of Γ under this quotient map is C-dense in I/I_{0}. As both groups are discrete, this clearly implies that the inclusion Γ/Γ_{0} ⟶ I/I_{0} is of finite index. Thus the claim is proved.

Now note that Fr(X)/I is clearly compact, and is a manifold since I is acting freely and properly. Hence the leaf-space of F is also a compact manifold. Since each leaf of F is the inverse image of a point under the map from Fr(M) to the leaf space, we have that each leaf of F is compact. ⋄

It will be useful to know that I_{0} cannot have compact factors.

Claim II: I_{0} has no nontrivial compact factor.

Proof. In proving this claim, we will use degree theory for noncompact manifolds, phrased in terms of locally finite homology H_{∗}^{lf} (see, e.g., [Iv] for a discussion). Locally finite homology is the theory of cycles which pair with cohomology with compact support. Perhaps the quickest description of H_{∗}^{lf}(X) is as the usual reduced homology H̃_{∗}(X̂) of the one-point compactification X̂ of X. Alternatively, it can be described (for locally finite simplicial complexes) as the homology of the chain complex of infinite formal combinations of simplices for which only finitely many simplices with nonzero coefficients intersect any given compact region.

With this definition, it is easliy verified (see [Iv]) that the usual degree theory holds for continuous quasi-isometries between (possibly noncompact) manifolds X, with the fundamental class of the n-manifold X now being an element of H_{n}^{lf}(X, Z). As one example, the universal cover X of a closed, aspherical n-manifold M has a nonzero fundamental class lying in H_{n}^{lf}(X, Z). With this degree theory in place, we can now begin the proof of Claim II.

Now suppose that I_{0} has a nontrivial compact factor K. Since I_{0} is connected and dim(I_{0}) > 0 by assumption, we have that K is connected and dim(K) > 0.

Since M is closed, and so there is a compact fundamental domain for the Γ-action on X, we easily see that there exists a constant C so that each K-orbit has diameter at most C. But then X/K is quasi-isometric to X. Now the standard “connect the dots” trick (see, e.g., p.527 of [BW], or Appendix A of [BF] for exact details) states that such quasiisometries are a bounded distance (in the sup norm) from a continuous quasi-isometry (i.e. Lipschitz map). Hence there are continuous maps X ⟶ X/K and X/K ⟶ X inducing the given quasi-isometry. Since dim(K) > 0 we have that dim(X/K) < dim(X) = n. This implies that the fundamental class of X in H_{n}^{lf}(X, R), where n = dim(X), must vanish, contradicting the fact that X is the universal cover of a closed, aspherical n-manifold. ⋄

The next step in our proof of Theorem 1.2 is to determine information which will help us construct the orbifold base space B of the orbibundle (1.1).

Claim III: X/I_{0} is contractible.

Proof. The Conner Conjecture, proved by Oliver [Ol], gives that the quotient of a contractible manifold by a connected, compact, smooth transformation group is contractible. Our claim that X/I_{0} is contractible follows directly from the following simple extension of Oliver’s theorem.

Proposition 2.2. Let G be a connected Lie group acting properly by diffeomorphisms on a contractible manifold X. Then the underlying topological space of the orbit space X/G is contractible.

Proposition 2.2 is a consequence of Oliver’s Theorem and the following.

Proposition 2.3. Let G be a connected Lie group acting properly by diffeomorphisms on an aspherical manifold X. Denote by K the maximal compact subgroup of G. Then there exists an aspherical manifold Y such that X is diffeomorphic to Y × G/K, the manifold Y has a K-action, and the original action is given by the product action. In particular, X/G is diffeomorphic to Y/K.

Proof. Let \underline{EG} be the classifying space for proper CW G-complexes, so that \underline{EG}/G is the classifying space for proper G-bundles (see, e.g., the appendix of [BCH]). Now G/K is an \underline{EG} space. Hence there is a proper G-map ψ : X ⟶ \underline{EG}. But \underline{EG} has only one G-orbit, so ψ is surjective. Now let Y = ψ^{−1}([K]), where [K] denotes the identity coset of K. Hence X is diffeomorphic to G ×_{K} Y, and we are done.

⋄

We are now ready to construct, on the level of universal covers, the orbibundle (1.1), and in particular to prove that the base space B is a Riemannian orbifold. The crucial point is to understand stabilizers of the I_{0} action on X. For x ∈ X, denote the stabilizer of x under the I_{0} action by I_{x} := {g ∈ I_{0} : gx = x}. Let K_{0} denote the maximal compact subgroup of I_{0}; it is unique up to conjugacy.

Claim IV: I_{x} = K_{0} for each x ∈ X. Hence the following hold:

1. X/I_{0} is a manifold.

2. Each I_{0}-orbit in X is isometric to the contractible, homogeneous manifold I_{0}/K_{0}, endowed with some left-invariant Riemannian metric.

3. The natural quotient map gives a Riemannian fibration

I_{0}/K_{0} ⟶ X ⟶ X/I_{0}  (2.3)

Proof. Clearly I_{x} ⊆ K_{0}. Iwasawa proved ([Iw], Theorem 6) that any maximal compact subgroup of a connected Lie group is connected. Hence it is enough to prove that dim(I_{x}) = dim(K_{0}).

To this end we consider rational cohomological dimension cd_{Q}. By Claim III we have X/I_{0} is contractible. Since Γ/Γ_{0} acts properly on X/I_{0}, we then have

cd_{Q}(Γ/Γ_{0}) ≤ dim(X/I_{0})  (2.4)

Since K_{0} is maximal, we know I_{0}/K_{0} is contractible. By Claim I, we have that Γ_{0} is a uniform lattice in I_{0}, and so

cd_{Q}(Γ_{0}) = dim(I_{0}/K_{0})  (2.5)

Since X is contractible and M = X/Γ is a closed manifold, by general facts about cohomological dimension (see [Bro], Chapter VIII (2.4)), we have

dim(X) = cd_{Q}(Γ) ≤ cd_{Q}(Γ_{0}) + cd_{Q}(Γ/Γ_{0})

which combined with (2.4) and (2.5) gives

dim(X) ≤ dim(X/I_{0}) + dim(I_{0}/K_{0})  (2.6)

But for each x ∈ X, we have

dim(X) ≥ dim(X/I_{0}) + dim(I_{0}/I_{x})  (2.7)

which combined with (2.6) gives

dim(I_{0}/I_{x}) ≤ dim(I_{0}/K_{0})

and so dim(I_{x}) ≥ dim(K_{0}), as desired. Thus I_{x} = K_{0}.

It follows that each orbit I_{0} · x is diffeomorphic to a common Euclidean space I_{0}/K_{0}, so by the Slice Theorem (see, e.g., [Br], Chap. IV, §3,4,5) it follows that X/I_{0} is a manifold. We note that while the Slice Theorem is usually stated for actions of compact groups, the proof extends immediately to the case of proper actions of noncompact groups; one simply produces an invariant Riemannian metric by translating a compactly supported pseudometric, and this gives the required structure via exponentiation. ⋄

Finishing the proof. The action of Γ on X induces actions of Γ_{0} on I_{0}/K_{0}, and of Γ/Γ_{0} on X/I_{0}, compatible with the Riemannian fibration (2.3). By Myers-Steenrod, Γ/Γ_{0} acts properly discontinuously on X/I_{0}; we denote the quotient space of this action by B. We thus have a Riemannian orbibundle (as defined in the introduction):

F ⟶ M ⟶ B  (2.8)

where F denotes the closed, locally homogeneous Riemannian manifold Γ_{0}\I_{0}/K_{0}, endowed with the quotient metric of a left I_{0}-invariant metric on I_{0}/K_{0}. This completes the proof of Theorem 1.2

3 The case when I_{0} is semisimple

The main goal of this section is to prove Proposition 3.1 below, which shows that when I_{0} is semisimple with finite center, a much stronger conclusion holds in Theorem 1.2.

Proposition 3.1. Suppose that I_{0} is semisimple with finite center. Then M has a finite cover which is a Riemannian warped product N ×B, where N is nonempty, locally symmetric with nonpositive curvature, and has no local torus factors. In particular, π_{1}(B) ◁ π_{1}(M), and any nontrivial, normal abelian subgroup of π_{1}(M) lies in π_{1}(B).

Remark on semisimplicity. We would like to emphasize that by calling a connected Lie group G “semisimple” we mean only that the Lie algebra of G is semisimple. Thus the center Z(G) may be infinite. Such examples do exist (for example the universal cover of U(n, 1)) , and must be taken into account. We also point out that G may in general have compact factors. For the connected component I_{0} of the isometry group of the universal cover of a closed, aspherical Riemannian manifold, however, we have already proven in Claim II of §2 above that I_{0} has no nontrivial compact factor. Even so, the semisimple part (I_{0})^{ss} may have nontrivial compact factors coming from Z(I_{0}).

After proving Proposition 3.1, we show in §3.2 that the hypothesis that I_{0} is semisimple with finite center is more common than one might guess. Indeed, in Proposition 3.3 we prove that I_{0} is always semisimple with finite center unless Γ = π_{1}(M) contains an infinite, normal abelian subgroup.

3.1 The proof of Proposition 3.1

The structure of the proof of Proposition 3.1 is to first prove it at the level of fundamental groups, mostly using Lie theory. The theory of harmonic maps, as well as the existence of arithmetic lattices, is then used to build so many isometries of the universal cover of M that it is forced to fiber in the claimed way.

Triviality of the extension. Our first goal will be to prove that, after replacing Γ by a finite index subgroup if necessary, the exact sequence

1 ⟶ Γ_{0} ⟶ Γ ⟶ Γ/Γ_{0} ⟶ 1  (3.1)

splits as a direct product. As with every extension, (3.1) is determined by two pieces of data:

1. A representation ρ : Γ/Γ_{0} ⟶ Out(Γ_{0}), and

2. A cohomology class in H^{2}(Γ/Γ_{0}, Z(Γ_{0})_{ρ}), where Z(Γ_{0})_{ρ} is a Γ/Γ_{0}-module via ρ.

We analyze these pieces in turns. Let < I_{0}, Γ > be the smallest subgroup of I containing I_{0} and Γ. Consider the exact sequence

1 ⟶ I_{0} ⟶< I_{0}, Γ >⟶ Γ/Γ_{0} ⟶ 1  (3.2)

and let

ρ_{1} : Γ/Γ_{0} ⟶ Out(I_{0})

denote the induced action; this is just the action induced by the conjugation action of Γ on I. Since I_{0} is semisimple, we know (see, e.g. [He], Theorem IX.5.4) that Out(I_{0}) is finite. Hence, after passing to a finite index subgroup of Γ if necessary, we may assume that ρ_{1} is trivial. In other words, the Γ-action on I_{0} is by inner automorphisms, giving a representation

ρ_{2} : Γ/Γ_{0} ⟶ I_{0}/Z(I_{0})

Now, the conjugation action of Γ on I_{0} preserves Γ_{0}, and so the image of ρ_{2} lies in the normalizer N_{H}(Γ_{0}) of Γ_{0} in H := I_{0}/Z(I_{0}). Note that Γ_{0} ∩ Z(I_{0}) is finite and hence trivial, as is Z(Γ_{0}), since Γ_{0} is torsion free, and so Γ_{0} can be viewed as a subgroup of H. Since H is semisimple and Γ_{0} is a cocompact lattice in H (by Claim I in the proof of Theorem 1.2), it follows that N_{H}(Γ_{0})/Γ_{0} is finite ^{3}.

Hence, by replacing Γ with a finite index subgroup if necessary, we may assume ρ_{2} has trivial image. We thus have that the conjugation action of Γ on Γ_{0} is by inner automorphisms of Γ_{0}. Since Z(Γ_{0}) is trivial, the representation ρ : Γ/Γ_{0} ⟶ Out(Γ_{0}) is trivial ^{4}. We also know that

H^{2}(Γ/Γ_{0}, Z(Γ_{0})_{ρ}) = 0

since Z(Γ_{0}) = 0. It follows that, up to finite index, the exact sequence (3.1) splits, and in fact that

Γ ≈ Γ_{0} × Γ/Γ_{0}  (3.3)

Recall (2.8), where we found a Riemannian orbibundle

F ⟶ M ⟶ B

Our goal now is to use (3.3) to find a section of this fibration, and to use this to prove that M is a Riemannian warped product. In order to do this we will use the following tool.

Harmonic maps. We recall that a map f : N ⟶ M between Riemannian manifolds is harmonic if it minimizes the energy functional

E(f) = ∫_{N} ||Df_{x}||^{2}dvol_{N}

The key properties of harmonic maps between closed Riemannian manifolds which we will need are the following (see, e.g. [SY]):

• (Eels-Sampson) When the target manifold has nonpositive sectional curvatures, a harmonic map exists in each homotopy class.

• (Hartman, Schoen-Yau) If a harmonic map f : M ⟶ N induces a surjection on π_{1}, and if π_{1}(N) is centerless, then f is unique in its homotopy class. This follows directly from Theorem 2 of [SY].

3This follows for example from Bochner’s classical result that the closed manifold M = Γ\H/K has finite isometry group since it has negative Ricci curvature, and Isom(M) = N_{H}(Γ_{0}). For another proof, see [Ma], II.6.3.

4Note that there are cases when Out(Γ_{0}) is nontrivial; for example when Γ_{0} is a surface group then Out(Γ_{0}) is the mapping class group of that surface.

• (easy) The precomposition and postcomposition of a harmonic map with an isometry gives a harmonic map.

Showing that X is a warped product. The isomorphism in (3.3) gives via projection to a direct factor a natural surjective homomorphism π : Γ ⟶ Γ_{0}. Recall that 0 = Z(Γ_{0}) ⊇ Z(I_{0}) ∩ Γ_{0}, and so the injection Γ_{0} ⟶ I_{0} gives an injection Γ_{0} ⟶ I_{0}/Z(I_{0}). Our first goal is to extend the projection π to a projection π̂ :< I_{0}, Γ >⟶ I_{0}/Z(I_{0}).

To this end, note that Z(I_{0}) is characteristic in I_{0}, and so Z(I_{0}) ◁ I; in particular, Z(I_{0})◁ < I_{0}, Γ >. Taking the quotient of the exact sequence (3.2) by the finite normal subgroup Z(I_{0}) gives an exact sequence

1 ⟶ I_{0}/Z(I_{0}) ⟶< I_{0}, Γ > /Z(I_{0}) ⟶ Γ/Γ_{0} ⟶ 1  (3.4)

We claim that the kernel of (3.4) is centerless. Indeed, if G is any connected semisimple Lie group, then its center Z(G) is clearly closed, hence discrete since G is semsimple. But for any connected Lie group G with Z(G) discrete, the center of G/Z(G) is trivial (see, e.g., Exercise 7.11(b) of [FH]). The reason this fact is true can be seen from the fact that the discreteness of Z(G) implies that the quotient map G ⟶ G/Z(G) is a covering map of Lie groups, and so both G and G/Z(G) have isomorphic Lie algebras and isomorphic universal covers..

Since the kernel of (3.4) is centerless, the exact argument as above gives that (3.4) splits, so that

< I_{0}, Γ > /Z(I_{0}) ≈ I_{0}/Z(I_{0}) × Γ/Γ_{0}  (3.5)

This isomorphism, composed with the natural projections, then gives us a surjective homomorphism

π̂ :< I_{0}, Γ >⟶ I_{0}/Z(I_{0})

Let K_{0} denote a maximal compact subgroup of the semisimple Lie group I_{0}. We then have that I_{0} acts isometrically on the contractible, nonpositively curved symmetric space of noncompact type X_{0} := I_{0}/K_{0}. Since Z(I_{0}) is finite, it lies in K_{0}, and so the I_{0} action on X_{0} factors through a faithful action of I_{0}/Z(I_{0}). As X_{0} is contractible, the homomorphism π is induced by some continuous map h : X/Γ ⟶ X_{0}/Γ_{0}. Thus f is homotopic to a harmonic map h. By the theorem of Hartman and Schoen-Yau stated above, f is the unique harmonic map in its homotopy class.

Claim 3.2. The lifted map f̃: X ⟶ X_{0} is equivariant with respect to the representation π̂ :< I_{0}, Γ >⟶ I_{0}/Z(I_{0}).

To prove this claim, first note that f̃ is equivariant with respect to the representation π, by construction; we want to promote this to π̂-equivariance. One strange aspect of this is that we use an auxilliary arithmetic group, which seems to have nothing to do with the situation.

To begin, consider any cocompact lattice Δ in I_{0}/Z(I_{0}). By (3.5), Δ × Γ/Γ_{0} is a cocompact lattice in < I_{0}, Γ > /Z(I_{0}), so it pulls back under the natural quotient to a cocompact lattice, which we will also call Δ, in < I_{0}, Γ > (recall that Z(I_{0}) is finite).

Then, up to translation by elements of Δ, there is a unique harmonic map ϕ_{Δ} : X ⟶ X_{0} equivariant with respect to the restriction π̂|_{Δ×(Γ/Γ_{0})}. Suppose Δ′ is any other lattice in I_{0} which is commensurable with Δ. Since both ϕ_{Δ} and ϕ_{Δ′} are harmonic and equivariant with respect to the representation π̂ restricted to (Δ ∩ Δ′) × (Γ/Γ_{0}), and since Δ ∩ Δ′ has finite index in both Δ and in Δ′, we have by uniqueness of harmonic maps that ϕ_{Δ} = ϕ_{Δ′}. We remark that this “uniqueness implies equivariance” principle is also a key trick in [FW].

Since I_{0} is semisimple with finite center, the quotient I_{0}/Z(I_{0}) is semisimple and centerless (as proven just after equation (eq:ss4b) on page 15 above). By a theorem of Borel ([Bo], Theorem C), there exists a cocompact arithmetic lattice Δ_{1} in I_{0}/Z(I_{0}). Since I_{0}/Z(I_{0}) is centerless, it follows that the commensurator Comm_{I_{0}/Z(I_{0})}(Δ_{1}) is dense in I_{0}/Z(I_{0}); see, for example, Proposition 6.2.4 of [Zi], where this is clearly explained.

Let Δ_{0} denote the pullback of Δ_{1} under the natural quotient map I_{0} ⟶ I_{0}/Z(I_{0}). Since Δ_{0} contains Z(I_{0}), and so Comm_{I_{0}}(Δ_{0}) is the central extension of Comm_{I_{0}/Z(I_{0})}(Δ_{1}) associated to Z(I_{0}), it follows that Comm_{I_{0}}(Δ_{0}) is dense in I_{0}. At this point, a verbatim application of the proof of the “arithmetic case” of Theorem 1.4 in [FW] completes the proof of Claim 3.2; for completeness, we briefly recall this proof.

Let U denote the set of g ∈< I_{0}, Γ > for which the equation

ϕ_{Δ_{0}}g = gϕ_{Δ_{0}}  (3.6)

holds. Now U is closed, and the uniqueness of harmonic maps gives that U is a subgroup of I_{0}. Hence U is a Lie subgroup of I_{0}. Applying the above paragraphs with Δ = Δ_{0} and with Δ′ running through the collection L of lattices commensurable with Δ_{0} in I_{0}, gives that U contains every lattice in L. Since Comm_{I_{o}}(Δ_{0}) is dense in I_{0}, there are infinitely many distinct members of L conjugate to Δ_{0}, namely the conjugates of Δ_{0} by elements of Comm_{I_{o}}(Δ_{0}). Hence U is nondiscrete, hence positive dimensional. Under the adjoint representation, Δ_{0} preserves the Lie algebra of U. But Δ_{0} is a lattice in I_{0}, hence is Zariski dense by the Borel Density Theorem (see, e.g. [Ma], Theorem II.2.5). Thus U = I_{0}, finishing the proof of Claim 3.2.

We now have a map

X/(Γ × (Γ/Γ_{0})) ⟶ (X_{0}/Γ_{0}) × (X/I_{0})/(Γ/Γ_{0})

given by the product of f̃ and the natural orbit map. This map harmonic when composed with projection to the first factor, and is clearly a diffeomorphism, since we have just shown that the first coordinate is equivariant with respect to π̂.

3.2 Consequences of no normal abelian subgroups

The assumption that Γ = π_{1}(M) contains no nontrivial normal abelian subgroup has strong consequences for our setup. The main one is the following.

Proposition 3.3. Suppose Γ contains no infinite, normal abelian subgroup. Then I_{0} is semisimple with finite center.

Proof. Note that since Γ is torsion free, it follows that Γ has no normal abelian subgroups; in particular Z(Γ) = 1.

For any connected Lie group G there is an exact sequence

1 ⟶ G^{sol} ⟶ G ⟶ G^{ss} ⟶ 1  (3.7)

where G^{sol} denotes the solvable radical of G (i.e. the maximal connected, normal, solvable Lie subgroup of G), and where G^{ss} is the connected semisimple Lie group G/G^{sol}.

Let Γ^{sol} denote the maximal normal solvable subgroup of Γ; it is of course torsion free since Γ is torsion free. We claim that Γ^{sol} is trivial. Suppose not. Being a nontrivial torsionfree solvable group, Γ^{sol} would then have an infinite, characteristic, torsion-free abelian subgroup H, namely the last nontrivial term in its derived series. Since H is characteristic in the normal subgroup Γ^{sol} of Γ, it would follow that H is normal in Γ. Since H is infinite abelian, this contradicts the hypothesis on Γ.

We now quote a result of Prasad, namely Lemma 6 in [Pr]. For a lattice Γ in a Lie group I_{0}, Conclusion (2) of Prasad’s Lemma gives, in the terminology of [Pr]:

rank(Γ^{sol}) = χ(I_{0}^{sol}) + rank(Z(I_{0}^{ss}))

Here χ(I_{0}^{sol}) denotes the dimension of I_{0}^{sol} minus that of its maximal compact subgroup, rank(Z(I_{0}^{ss})) denotes the the rank of the center of I_{0}^{ss}, and rank denotes the sum of the ranks of the abelian quotients in the derived series. Since in our case we have proven that Γ^{sol} = 0, it follows both that χ(I_{0}^{sol}) = 0, i.e. that I_{0}^{sol} is compact, and that the rank of Z(I_{0}) is 0, so that Z(I_{0}) is finite.

Since I_{0}^{sol} is both solvable and compact, it is a torus T. Since the automorphism group of T is discrete (namely it is GL(dim(T), Z)), the natural conjugation action of the connected group I_{0}^{ss} on T given by (3.7) must be trivial, so that T is a direct factor of I_{0}. But we have already proven (Claim II of §2) that I_{0} has no nontrivial compact factors, a contradiction unless T is trivial. Thus I_{0}^{sol} = T is trivial; that is, I_{0} is semisimple. ⋄

Remark. It is possible to weaken the hypothesis of Proposition 3.3, and hence of all of the results which rely on it, to assuming only that Γ contains no finitely generated, infinite normal abelian subgroups. To do this, we begin by recalling that Prasad’s result used above also gives that the group Γ^{sol} is a lattice in some connected solvable subgroup S of I_{0}. It follows from Proposition 3.4 below that Γ^{sol} is polycyclic. But it is well-known and easy to see that any polycyclic group has the property that each of its subgroups is finitely generated (see, e.g. [Ra], Prop. 3.8). Hence the subgroup H constructed in the proof of Proposition 3.3 would in fact be finitely generated.

In the argument just given we needed the following proposition, proved by Mostow in the simply connected case.

Proposition 3.4. Every lattice Λ in a connected solvable Lie group S is polycyclic.

Proof. First note that π_{1}(S) is finitely-generated and abelian, and so the universal cover S̃ is a central Z^{d} extension of S for some d ≥ 0. The lattice Λ pulls back to a lattice Λ̃ in S̃, which is a central Z^{d} extension of Λ. Mostow proved (see, e.g. [Ra], Prop. 3.7) that any lattice Λ̃ in a connected, simply-connected solvable Lie group S̃ must be polycyclic. It follows easily that Λ is polycyclic. ⋄

The use of Prasad’s result simplifies the approach to Proposition 3.3 given in an earlier version of this paper. As part of that earlier approach, we proved the following proposition. We include this result here since we believe it might prove useful in the future, since the proof is direct, and since we were not able to find this result in the literature. The argument was kindly supplied to us by the referee.

Proposition 3.5. Let G be a connected semsimple Lie group, and let Λ be a lattice in G. If Z(G) is infinite then Z(Λ) is infinite.

Proof. Let T be the identity component of the closure of Z(G)Λ in G. First note that T is abelian; indeed, the commutator subgroup [T, T] of T is contained in the closure of the subgroup

[Z(G)Λ, Z(G)Λ] = [Λ, Λ] ⊂ Λ

and hence [T, T], being connected, is trivial.

Now let C be the unique maximal compact, connected normal subgroup of G. Then the Borel Density Theorem applied to the image of Z(G)Λ in G/C gives that the image of T in G/C is a connected, normal abelian subgroup. Hence it must be trivial. Thus T ⊆ C, and so it is a torus normalized by Z(G)Λ, and T Z(G)Λ is a closed subgroup of G containing the lattice Λ. Thus T Z(G)Λ/Λ has finite volume, which in turn implies that T Z(G) ∩ Λ is a lattice in T Z(G). Since Z(G) is infinite by hypothesis, and since T is a torus, we conclude that Λ′ := T Z(G) ∩ Λ is an infinite normal abelian subgroup of Λ. Since [Λ, Λ′] ⊂ [Λ, T] ⊂ T, and since T is compact, we have that [Λ, Λ′] is finite. Now since Λ is finitely generated (every lattice in a connected Lie group is finitely generated), we can conclude easily that a subgroup of Λ′ of finite index is contained in Z(Λ). This proves that Z(Λ) is infinite. ⋄

4 Some applications

In this section we finish the proof of Theorem 1.3. We then use Theorem 1.2 and its proof, and also Theorem 1.3, to prove the other theorems and corollaries stated in the introduction.

4.1 No normal abelian subgroups (proof of Theorem 1.3)

The fact that (2) implies (1) follows immediately from well-known properties of closed, locally symmetric Riemannian manifolds. Such M are aspherical by the Cartan-Hadamard theorem. Any normal abelian subgroup is trivial since the symmetric space M̃ has no Euclidean factors. The other two properties follow from the definitions.

To prove that (1) implies (2), we first quote Proposition 3.3 followed by Proposition 3.1. This gives that M has a finite-sheeted Riemannian cover M′ of M which is a smooth (indeed Riemannian warped) product M′ = N × B, where N is is isometric to a nonempty, irreducible, locally symmetric, nonpositively curved manifold. But M′ is smoothly irreducible by hypothesis, so that B must be a single point. It follows that M′ = N is locally symmetric. Since the metric on M′ was lifted from M, we have that M is locally symmetric.

4.2 Word-hyperbolic groups (proof of Corollary 1.4)

Again, (2) implies (1) follows immediately from the basic properties of closed, rank one locally symmetric manifolds.

To prove that (1) implies (2), first note that no torsion-free word-hyperbolic group can virtually be a nontrivial product, since then it would contain a copy of Z × Z. It then follows from Theorem 1.3 that M is locally symmetric. But every closed, locally symmetric manifold M either contains Z×Z in its fundamental group, or M must be negatively curved; hence the latter must hold for M.

4.3 Almost simple groups (proof of Corollary 1.5)

This follows just as the proof of Corollary 1.4, but using the following fact: an irreducible, cocompact lattice in a noncompact semisimple Lie group G is almost simple if and only if rankR ≥ 2. The “if” direction is the statement of the Margulis Normal Subgroup Theorem (see [Ma], Thm. IX.5.4). For the “only if” direction, first recall that cocompact lattices in rank one semisimple Lie groups are non-elementary word-hyperbolic. Such groups are never almost simple; for example, a theorem of Gromov-Olshanskii (see [Ol]) gives that all such groups have infinite torsion quotients.

4.4 Universal bound (proof of Theorem 1.7)

A theorem of Kazhdan-Margulis (see, e.g., [Ra], Corollary XI.11.9) shows that, for every connected semisimple Lie group G, there exists ϵ = ϵ(G) such that the covolume of every lattice in G is greater than ϵ. Let ϵ be this constant for G = Isom(M̃). Now let C_{1} = Vol(M, g_{0})/ϵ.

Now let X denote M̃ endowed with any fixed Riemannian metric h lifted from M. If Isom(X) is not discrete, then by Theorem 1.3 we have that h is homothetic to g_{0}, and we are done. If Isom(X) is discrete, then X/ Isom(X) is a compact orbifold, Riemannian covered by the compact manifold M, with degree of the cover d := [Isom(X) : π_{1}(M)]. Since the volume of a cover is multiplicative in degree, we have that

Vol(M, h) = d Vol(X/ Isom(X))) > dϵ

Now, Gromov has shown (see [Gr], and also [CF] for a different, and more detailed, proof) that the “minvol” invariant of M is positive; this means that there is a constant C_{2} = C_{2}(M) so that Vol(M, h) ≤ C_{2} Vol(M, g_{0}) for all h with sectional curvatures bounded above by 1 in absolute value. After rescaling, we can assume the given h satisfies this curvature bound. We then have

d < Vol(M, h)/ϵ ≤ C_{2} Vol(M, g_{0})/ϵ = C_{1}C_{2}

and we are done after setting C = C_{1}C_{2}.

4.5 Models for compact and finite volume manifolds (proof of Theorem 1.8)

Let Γ_{1} (resp. Γ_{2}) be a cocompact (resp. noncocompact) lattice in Isom(X). Since Isom(X) contains Γ_{1}, it acts cocompactly on X. Suppose Isom(X) were discrete, so that Isom(X) acts properly and cocompactly on X. Then the (orbifold) quotient Isom(X)\X would have finite volume. Since covolume is multiplicative in index, it would follow that Γ_{2} < Isom(X) has finite index. But then Γ_{2} would act cocompactly since Isom(X) does, a contradiction. Hence Isom(X) is not discrete and we may apply Theorem 1.2.

We thus obtain a Riemannian orbibundle, which at the level of universal covers gives a Riemannian warped product structure Y ˜×X_{0}, with Y the universal cover of B, where the metric has the property that for each x ∈ X, the metric on x × X_{0} is an I_{0}- homogeneous metric, depending on x.

Let Λ_{i} := Γ_{i}∩I_{0}, i = 1, 2. Claim I of the proof of Theorem 1.2 gives that Λ_{1} is cocompact. We must now prove that Λ_{2} is a noncocompact lattice.

To this end, first note that π := Γ_{2}/Λ_{2} is a cocompact lattice in Isom(Y). First suppose B is not 1-dimensional. We can then perturb the metric on B to get a new universal (in the category of orbifolds) cover Y′ with Isom(X) = Isom(Y′×X_{0}) but with Isom(Y′) = π_{1}^{orb}(B). Now π is a lattice in Isom(Y′), so it has finite index in π_{1}(B).

We thus have that each of Γ_{i}, i = 1, 2 can be written as a group extension with kernel Λ_{i} and quotient a group with the same rational cohomological dimension cdQ as π_{1}(B). We consider rational cohomological dimension cdQ in order to deal with the fact that π_{1}(B) might not be virtually torsion-free.

Since each Γ_{i} acts properly on the contractible manifold X, and since Γ_{1} acts cocompactly and Γ_{2} does not, we have that cdQ(Γ_{2}) < cdQ(Γ_{1}) (see, e.g. [Bro], VIII.8).

Now, if Λ_{2} were cocompact, then Γ_{2} would be an extension of fundamental groups of closed, aspherical manifolds, and so cdQ(Γ_{2}) would be the sum of the cdQ of the kernel and quotient; but this sum equals cdQ(Γ_{1}) (see, e.g., Theorem 5.5 of [Bi]), a contradiction. Hence Λ_{2} is not cocompact.

If B is one-dimensional, then no perturbation as above exists. To remedy this, we simply take the product of B with a closed, genus 2 surface, endowed with a Riemannian metric with trivial isometry group. We then run the rest of the argument verbatim.

4.6 Irreducible lattices (proof of Theorem 1.9)

We first note that the hypothesis implies Isom(X) is not discrete. Since in addition X has a cocompact discrete subgroup, we may apply Theorem 1.2. Hence X is isometric to a warped Riemannian product X = Y ˜×X_{0}. By the proof of Theorem 1.2, the group Isom(X_{0}) corresponds with connected component of the identity of Isom(X). In particular Isom(Y) must be discrete. The theorem follows easily.

4.7 The Hopf Conjecture

A well-known conjecture of Hopf-Chern-Thurston states that the Euler characteristic of any closed, aspherical manifold M^{2k} satisfies (−1)^{k}χ(M^{2k}) ≥ 0. A stronger conjecture of Singer posits that the L^{2}-cohomology of M^{2k} vanishes except in dimension k (see, e.g. [CG, Lu]). These conjectures are completely open except when k = 1.

We will now prove that the Hopf Conjecture holds for those smooth, aspherical manifolds M^{2k} which admit some Riemannian metric with symmetry.

Theorem 4.1. Let M^{2k} be any closed, aspherical, smooth manifold which is smoothly irreducible. If M admits some Riemannian metric so that the induced metric on the universal cover M̃ satisfies [Isom(M̃) : π_{1}(M)] = ∞, then the Singer Conjecture (and hence the Hopf Conjecture) is true for M^{2k}.

The Singer Conjecture clearly holds for products of surfaces and also for products of any 3-manifolds with S^{1}. Thus Theorem 4.1 holds in dimension four without the assumption that M^{4} is smoothly irreducible.

Proof of Theorem 4.1. If π_{1}(M^{2k}) has no nontrivial normal abelian subgroups and is smoothly irreducible, then by Theorem 1.3 it admits a locally symmetric Riemannian metric. The Singer Conjecture is known for such manifolds (see, e.g. [Lu], Cor. 5.16). If π_{1}(M^{2k}) does contain a nontrivial, normal abelian subgroup A, then one may apply a theorem of Cheeger-Gromov ([CG], Cor. 0.6) which gives, even more generally for amenable A, that Singer’s Conjecture holds. ⋄

4.8 Complex manifolds (proof of Theorem 1.10)

As pointed out above, Theorem 1.10 follows immediately from Proposition 0.1 in [Na] together with Claim IV in the proof of Theorem 1.2.

Actually, we can do without Nadel’s result under certain mild assumptions. Since Aut(M̃) acts isometrically in some Riemannian metric, we can apply Theorem 1.2 directly to obtain the claimed splitting, but with two problems: first, the splitting is an isometric (not holomorphic) one; and second, the factor M_{1} is only locally homogeneous, not necessarily locally symmetric.

The second problem is corrected once we know that Aut(M̃)^{0} is semisimple with finite center; equivalently, M_{1} doesn’t fiber with nontrivial solvmanifold fiber. One way to rule this out is to assume that π_{1}(M) contains no infinite normal abelian subgroups, giving us a new proof of Theorem 1.10 in this case. In complex dimension two, M_{1} must be locally symmetric, for otherwise it would have a fibering with torus fiber, giving that χ(M) = 0. However, in complex dimension two, we have

χ(M) = c_{2}(M) ≥ \frac{1}{3}c_{1}(M)^{2} > 0

by the Bogomolov-Miyaoka-Yau inequality.

To correct the first problem, once we have that the factor M_{1} is locally symmetric, we apply Siu’s Rigidity Theorem (also used in [Fr2]) to obtain that M_{1} is biholomorphic to a closed, Hermitian locally symmetric space.

5 An extension to the non-aspherical case

A number of the results from this paper generalize from closed, aspherical Riemannian manifolds to all closed Riemannian manifolds. As an illustrative example we give the following theorem. As far as we know, this is the first geometric rigidity theorem for non-aspherical manifolds with infinite fundamental group.

We say that the universal cover M̃ of a Riemannian manifold M has essential extra symmetry if, for a compact subset K ⊂ M̃, and for all ϵ > 0, there exists g ∈ Isom(M̃) such that • d(g(m), m) < ϵ for all m ∈ K

• sup d(g^{n}(m), m) = ∞ for any fixed m.

The condition that M̃ have an essential symmetry is equivalent to the identity component of Isom(M̃) being noncompact. If π_{1}(M) is torsion free, this is equivalent to Isom(M̃) not being an extension of π_{1}(M) by any compact group. (If the identity component of Isom(M̃) were compact, then π_{1}(M) would intersect this group in a normal lattice by Claim I, and this must be trivial if π_{1}(M) is torsion free.)

Theorem 5.1. Let M be any closed Riemannian manifold which, for simplicity, is smoothly irreducible. Suppose that π_{1}(M) contains no infinite, finitely-generated, normal abelian subgroup, and that M has essential extra symmetries. Then there exists a finite Riemannian cover M′ of M which is a fiber bundle over a closed, irreducible, locally symmetric manifold, with all fibers isometric. In particular the structure group of the bundle is compact.

The proof of Theorem 5.1 follows quite closely the proof of Theorem 1.3, with only a few adaptations. Hence instead of a detailed proof, we now just indicate the adaptations that are necessary.

Proof. We will describe how to modify the proof of Theorem 1.3 in order to prove Theorem 5.1. As in the proof of Theorem 1.3, the isometry group Isom(M̃) is a semidirect product of I_{0} and Δ, where Γ = π_{1}(M) intersects I_{0} in a lattice Γ_{0}. As before, the hypothesis that Γ has no infinite, finitely-generated, normal abelian subgroup implies that I_{0} is semisimple with finite center.

This implies as in §3.1 that, replacing M by a finite cover if necessary, we have a product structure on Γ. We then have a harmonic map f : M ⟶ K_{0}\I_{0}/Γ_{0}, where K_{0} is a maximal compact subgroup of I_{0}. This will provide the orbibundle structure on M, and the Lie group I_{0} will be responsible for the isometries among the fibers.

The proof of this is again based on the uniqueness of harmonic maps to nonpositively curved manifolds, when one has a surjection of fundamental groups, and no center in the image. Here, since Γ splits as a product, having such a center would give us a normal abelian subgroup, contradicting the hypothesis. On M̃ the Γ-equivariant map given by the lift of f extends, by the arithmetic group trick of Claim 3.2, to an I_{0}-equivariant map F : M̃ ⟶ I_{0}/K_{0}. Consequently the fibers are isometric to each other as the I_{0}-action on the target is transitive. ⋄

6 A truly singular orbibundle

In this section we construct a (7-dimensional) Riemmanian manifold M with the property that M is a Riemannian orbibundle, but no finite cover of M is a Riemannian fiber bundle.

M will also have the property that Isom(M̃) is not discrete. This will prove that the “orbibundle” conclusion of Theorem 1.2 cannot be improved to a fiber bundle, even if one is willing to pass to a finite cover.

We begin with the following construction, which we believe is of independent interest.

Theorem 6.1. There is a group Γ which acts cocompactly, properly discontinuously by diffeomorphisms on R^{n} for some n, but which is not virtually torsion free. Moreover, there exists ξ ∈ H^{2}(Γ, Z) which restricts to a nonzero class on some Z/pZ ∈ Γ, where p is a prime, and which also restricts to a nonzero class on every finite index subgroup.

Proof. Let G = Z/pZ ∗ Z. Then G acts properly discontinuously and cocompactly on its Bass-Serre tree T. In [Bri], Bridson shows that there is an amalgamated free product H = G∗_{F} G, with F a nonabelian free group, which has no finite quotients. Let Y be the universal cover of the standard Cayley 2-complex for H; hence H acts properly discontinuously and cocompactly on Y. It is easy to see that Y is contractible.

Equivariantly thicken Y (see [As] for the details of equivariant thickening). The action on this thickening gives a properly discontinuous, cocompact Γ-action on a contractbile 6dimensional manifold. Since the action is simplicial, we can perform the Davis reflection group construction (see [Da]) equivariantly to build a cocompact action of a group Λ on a contractible manifold, and with H being a retract of Λ. This action can in fact be made smooth, as is explained in §17 of [Da]. By a theorem of Stallings [St], the group Γ := Λ × Z then acts on R^{n} properly discontinuously and cocompactly by diffeomorphisms.

Note that any finite index subgroup of Γ intersects H in a finite index subgroup of H, which must therefore be all of H since H has no finite index subgroups. Hence the fixed Z/pZ subgroup of Γ must lie in each finite index subgroup of Γ.

Now by construction there is a surjection Γ ⟶ G. Let ξ ∈ H^{2}(Γ, Z) denote the pullback of the class generating H^{2}(G, Z). As the amalgamating subgroup F is free, the amalgamation dimension of H^{2}(H, Z) is at most one greater than H^{2}(G, Z). Since H is a retract of Λ, we know that ξ ∈ H^{2}(Λ, Z) is also nonzero. It follows easily that ξ pulls back to a nonzero class in Γ. ⋄

We now build the manifold M. Let Γ, ξ be given as in Theorem 6.1. Let Γ̃ denote the central extension of Γ given by the cocycle ξ ∈ H^{2}(Γ, Z). Since this cocycle vanishes in H^{2}(Γ, R), we have that Γ̃ lies in Γ × R. Now fix any Γ-invariant metric on R^{n}, and extend this to any (Γ × R)-invariant metric on R^{n} × R = R^{n+1}; call the resulting Riemannian manifold Y.

Now Γ × R acts properly discontinuously and cocompactly by isometries on Y ≈ R^{n+1}. The quotient M clearly satisfies the claimed properties. Note too that Isom(M̃) = Isom(Y) contains R, and so is not discrete.

Remark. in the examples above, the dimension of M is at least 7. We do not know whether this dimension can be lowered. Indeed, it seems difficult to obtain information about the geometry of such examples, although they do seem compatible with at least large-scale nonpositive curvature.

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Dept. of Mathematics, University of Chicago 5734 University Ave. Chicago, Il 60637 E-mail: farb@math.uchicago.edu, shmuel@math.uchicago.edu