7 Lp-cohomology of symmetric spaces ∗ 0 0 2 Pierre Pansu1,2 † n a February 2, 2008 J 5 ] G ABSTRACT.This is a shortsurveyofRiemanniangeometricapplica- D tions of Lp-cohomology of thick spaces, p=2. . 6 h t a 1 What is Lp-cohomology ? m [ 1.1 Definition 1 v Cohomology is the roughest invariant of topological spaces (much simpler 1 than the fundamental group, for instance). To take a metric into account, 5 one introduces decay conditions at infinity, this leads to Lp-cohomology. 1 1 0 Definition 1 Let M be a Riemannian manifold. Let p > 1. Let LpΩ (M) ∗ 7 denote the Banach space of Lp-differential forms and Ω ,p(M) the space of 0 ∗ / Lp-differential forms whose exterior differential is again Lp, equipped with h the norm t a m p p 1/p ω = ω + dω . : k kΩ∗,p k kLp k kLp v (cid:0) (cid:1) i The complex (Ω ,p(M),d) has a cohomology H ,p which is called the Lp- X ∗ ∗ cohomology of M. r a 1.2 Reduced cohomology and torsion When M is compact, the Lp condition is no restriction, Lp-cohomology coincides with usual de Rham cohomology. Therefore we shall concentrate on noncompact manifolds. ∗Keywords: Lp-cohomology,negativecurvature,symmetricspace,Besovspace. Math- ematics Subject Classification : 43A15, 43A80, 46E35, 53C20, 53C30, 58A14. †1 UnivParis-Sud, Laboratoire deMath´ematiques d’Orsay, Orsay,F-91405 ; 2 CNRS,Orsay, F-91405. 1 In general, Hk,p = (Ωk,p Kerd)/dΩk 1,p is not a Banach space, since − ∩ dΩk 1,p need not be closed. Therefore, one introduces a notation for its − Hausdorff quotient. Definition 2 Rk,p = (Ωk,p Kerd)/dΩk 1,p, − ∩ Tk,p = dΩk 1,p/dΩk 1,p. − − Rk,p, the reduced cohomology, is a Banach space. The torsion Tk,p does not have a norm. It is not a Hausdorff topological space. 1.3 Thick and thin ends For Riemannian manifolds withthin ends (e.g. Riemann surfaces with finite area cusps), one expects Lp-cohomology to coincide with the cohomology of some compactification, eventually with some small correction taking into account the specific metric behaviour of each end. This point of view has given riseto a hugelitterature. We refer to S.Zucker’s contribution to these proceedings. The present notes concentrate on Riemannian manifolds with bounded geometry, i.e. withsectionalcurvatureboundedonbothsidesandinjectivity radius bounded from below. For short, we call them thick spaces. Examples ofthickspacesarehomogeneousspacesofLiegroupsanduniversalcoverings of compact manifolds. We shall see that, for many thick spaces but not all, Lp-cohomology is again expressible in terms of some compactification, but the analytic (and not only topological) properties of the compactification play a role. 1.4 Example: The real hyperbolic plane H2 R HereH0,p = 0 = H2,p forall p. Letusbeginwithp = 2. SincetheLaplacian on L2 functions is bounded below, T1,2 = 0. Therefore H1,2 = R1,2 = L2 harmonic 1-forms { } = harmonic functions h on H2 with h L2 /R. R { ∇ ∈ } Since the Dirichlet integral h 2 in 2 dimensions is a conformal in- k∇ k variant, one can switch from thRe hyperbolic metric on the disk D to the euclidean metric on the disk. 2 Therefore H1,2 = harmonic functions h on D with h L2 /R { ∇ ∈ } = Fourier series Σa einθ with a = 0,Σ n a 2 < + n 0 n { | || | ∞} which is the Sobolev space H1/2(R/2πZ). More generally, for p > 1, T1,p = 0 and H1,p is equal to the Besov space B1/p(R/2πZ) mod constants. p,p Inthisexample,hyperbolicplaneiscompactifiedintoadisk,Lp-cohomology identifies with a function space on the boundary circle. 1.5 Example: the real line Inthatcase,H0,p = 0. R1,p = 0sinceeveryfunctioninLp(R)canbeapprox- imated in Lp with derivatives of compactly supported functions. Therefore H1,p is only torsion. It is non zero and thus infinite dimensional. Indeed, the 1-form dt (cut off near the origin) is in Lp for all p > 1 but it is not the t differential of a function in Lp. This is an example whereno compactification seems to help understand- ing Lp-cohomology. 1.6 Functoriality By definition, Lp-cohomology is obviously invariant under biLipschitz dif- feomorphisms. In the same way as cohomology is natural under continuous maps, and not only smooth maps, Lp-cohomology is natural under a wider class of maps, called uniform maps, for thick spaces. Say a map f :X Y → between metric spaces is uniform if d(f(x),f(x)) is bounded from above in ′ terms of d(x,x) only. Proving this requires a modification of the definition, ′ in order that it makes sense for rather general metric spaces. There are several possibilities, see [E2, F]. The following one is taken from [P4]. Definition 3 Let X be a metric space equipped with a measure. For each scale t > 0, consider the simplicial complex X whose vertices are points of t X and such that a subset ∆ X of k + 1 points forms the vertices of a ⊂ k-simplex if and only if its diameter is t. Simplicial cochains κ of X t ≤ possess an Lp norm 1/p κ = κ(x ,...,x )pdx ...dx . k kp (cid:18)Z | 0 k | 0 k(cid:19) X X ×···× This allows to define Lp-cohomology of X . t 3 Proposition 4 ([P4]). Let X be a Riemannian manifold admitting a co- compact isometric group action. Assume that Hj(X,R) = 0 for 1 j ≤ ≤ k. Then, for all t, Hk,p(X ) = Hk,p(X). Furthermore, Hk+1,p(X ) = t t ker(Hk+1,p(X) Hk+1(X,R)). → This means that Lp-cohomology depends only on the Riemannian mani- fold viewed as a coarse metric space. Theadvantage of definition 3 is that it applies todiscrete groupsaswell. Onemerely needsto select aleft-invariant metric whose balls are finite sets. Proposition 5 Let X, Y be measured metric spaces such that the volumes of balls are bounded above and below in terms of radius only. Then, for all k, for large enough t, there exists t > 0 such that f induces a map f : ′ ∗ Hk,p(Y ) Hk,p(X ). It follows that biuniform maps induce isomorphisms t t′ → on Lp-cohomology. Note that any homomorphism between discrete groups is uniform. In other words, Lp-cohomology is a functorial invariant of discrete groups which, in the case of uniform lattices in Lie groups, can probably be com- puted by analytic means. In this case, it is likely that the reduced part can be expressed in terms of some kind of boundary of the Lie group (com- pare A. Koranyi’s contribution to these proceedings), but torsion cannot be excluded. 2 What is it good for : case p = 2 L2 invariants constitute a rich theory by themselves, see [Lu2]. We extract only a few illustrations of the meaning and use of L2-cohomology, in con- nection with geometry, in the thick case. We will not develop these points, and refer to the litterature. 2.1 Square integrable harmonic forms If M is complete, Hodge theorem applies : Rk,2 identifies with the space of square integrable harmonic k-forms. Therefore, it seems to be computable by analytic means, as the example of hyperbolic plane shows. In return, L2-cohomology of real, complex and quaternionic hyperbolic spaces, and especially its description in terms of differential forms on the boundary, has been used to compute K-theoretic invariants of the groups SO(n,1), SU(n,1) and Sp(n,1), see [K, J1, JK, J2]. 4 2.2 Torsion in degree 1 Vanishing of T1,2(M) is equivalent to the following isoperimetric inequality. There exists a constant C such that for all smooth, compactly supported functions u on M, u C du . k kL2 ≤ k kL2 Under local bounded geometry assumptions, this is equivalent to the fol- lowing isoperimetric inequality. There exists a constant C such that for all smooth, compact domains D M, ⊂ vol(D) Cvol(∂D). ≤ When M covers a compact manifold N with fundamental group Γ, isoperi- metric inequality fails if and only if Γ is amenable, see [Gr]. Therefore amenable groups can be characterized by T1,2(M) = 0, [Br]. 6 ExamplesofamenablegroupsincludeZ(cf. therealline,above),solvable groups, groups of intermediate growth, see [GZ] for a state of the art. Examplesofnonamenablegroupsincludefreegroups,surfacegroups(cf. the hyperbolic plane above), nonelementary hyperbolic groups, lattices in Liegroups(exceptextensionsofsolvableLiegroupsbycompactLiegroups). 2.3 Group cohomology Assume that M is contractible and covers a compact manifold with fun- damental group Γ. Then H ,2(M) identifies with the cohomology of the ∗ regular representation of Γ. Certain classes of groups are known to have vanising cohomology in degree 1. For instance, a finitely generated group Γ isaKazhdan group ifandonlyifH1(Γ,π) = 0forallunitaryrepresentations π of Γ, see [DV]. For such a group, H1,2(M) = 0. Examples of Kazhdan groups include lattices in semi-simple Lie groups with no simple factors locally isomorphic to SO(n,1) or SU(m,1). Examples of non Kazhdan groups include amenable groups, free groups, surface groups and lattices in SO(n,1) or SU(m,1). 1. Amenable groups have T1,2 = 0 (see above) and R1,2 = 0 (see 2.5 6 below). 2. Free groups and surface groups have T1,2 = 0 and R1,2 = 0. 6 3. Lattices in SO(n,1) or SU(m,1) have H1,2 = 0 except for lattices in SO(2,1) = SU(1,1) which are surface groups. 5 2.4 Cohomology of towers of coverings Assumeagain that M is contractible and covers a compact manifold N with fundamental group Γ. If nonzero, Rk,2 is infinite dimensional. Neverthe- less, one can define a kind of dimension by unit volume dim (Hk,2) known Γ as the the k-th L2-Betti number bk,2(N) of N, [A]. This works for arbi- trary manifolds, but in case Γ can be exhausted by a tower of finite index normal subgroups Γ , the definition is easier : renormalized usual Betti j numbers b (Γ M)/[Γ : Γ ] converge to bk,2(N), [Lu1]. Thus R ,2 reflects k j j ∗ \ the behaviour of cohomology of large compact quotients. For expositions of L2-Betti numbers, see [Lu2] and [P3]. For a connection between L2-Betti numbers and spaces with thin ends, see the series of papers by J. Cheeger and M. Gromov, [CG2, CG3, CG4]. 2.5 Euler-Poincar´e characteristic of amenable groups It has been observed early that abelian or nilpotent groups have vanishing Euler-Poincar´e characteristic. It was not a trivial matter to extends this to solvable groups. In [CG1], J. Cheeger and M. Gromov have extended this vanishing theorem to the wide class of amenable groups. Their proof relies on L2-cohomology. The additivity of dim (it adds up exactly under direct Γ sums) implies that the Euler-Poincar´e characteristic of a group is equal to the alternating sum of its L2-Betti numbers. J. Cheeger and M. Gromov show that amenable groups have vanishing L2-Betti numbers. This follows easily from the isoperimetric characterization of amenable finitely generated groups. We cheat a bit : it is a delicate point to defineL2-Betti numbersfor arbitrary groups, notonly those whichadmit afinitedimensionalclassifying space. 2.6 Discrete series R ,2 is a Hilbert space on which the isometry group of M acts unitarily. ∗ In the case when M = G/K is a Riemannian symmetric space, Rk,2 splits as the direct sum of irreducible representations which belong to the discrete seriesandhavethesameinfinitesimalcharacter asthetrivialrepresentation, [Bo]. This provides us with a concrete realisation of these discrete series representations. Fix a uniform lattice Γ G. Each a discrete series representation π ∈ has a Harish Chandra formal dimension w(π), which is proportional to its 6 Γ-dimension, dim (π) Γ w(π) = const.(G) . vol(Γ G/K) \ L2-Betti numbers are thus proportional to sums of Harish Chandra formal dimensions. 2.7 Principal series Principalseriesrepresentationswhoseinfinitesimalcharacter isequaltothat of thetrivialrepresentation contributetothetorsionpartof L2-cohomology. Indeed, they are only weakly contained in the regular representation. This leads to the following theorem. Theorem 1 (A. Borel, [Bo]). Let M = G/K be a Riemannian symmetric space. 1. If G and K have equal ranks, then T ,2(M) = 0 and Rk,2(M) = 0 if ∗ 6 and only if dim(M) = 2k. 2. Otherwise, R ,2(M) = 0andTk,2(M) = 0ifandonlyif2k (dim(M) ∗ 6 ∈ − ℓ,dim(M)+ℓ], where ℓ = rankC(G) rankC(K). − This contribution to torsion can again bequantitatively measured. Back to the general case when M covers a compact manifold N with fundamental group Γ. All spectral projectors 1 (dd ) have finite Γ-traces, therefore (0,λ] ∗ one can define Novikov-Shubin numbers logtr 1 (dd ) Γ (0,λ] ∗ α (N) = limsup , k logλ λ 0 → which measure the polynomial decay of the spectral density function of dd ∗ on k-forms, [NS]. Theorem 2 (M. Olbrich, [O], N. Lohou´e, S. Mehdi [LM]). Let M = G/K be a Riemannian symmetric space. Assume ℓ = rankC(G) rankC(K) > 0. − Then, for all k (dim(M)−ℓ, dim(M)+ℓ], the k-th Novikov-Shubin invariant is ∈ 2 2 equal to α = ℓ. Otherwise, the L2-spectrum of dd is bounded below. k ∗ker(d∗)⊥ | 7 3 What is it good for : case p = 2 6 The theory for p = 2 is much less advanced. What is missing is a general- 6 ization of dim . Therefore no topological applications have been found yet. Γ Nevertheless, there exist significant geometric applications. They are less well known, therefore we shall develop them here. 3.1 Amenability again The statements of paragraph 2.2 extend if we replace 2 by any p > 1. Therefore a finitely generated group is amenable if and only if T1,p = 0 for 6 some (resp. all) p > 1. 3.2 Euler-Poincar´e characteristic of negatively curved man- ifolds A longstanding conjecture, attributed to H. Hopf, claims that the Euler- Poincar´e characteristic of a compact negatively curved 2m-manifold should be nonzero, and of the same sign as ( 1)m. J. Dodziuk and I. Singer have − proposed the following approach : prove that all L2-Betti numbers vanish, but the middle one, which is non zero, see [An, Y2]. This program has been completed yet only for manifolds which admit an auxiliary Ka¨hler metric, [G2], see section 4. 3.3 Hausdorff dimension at infinity In degree 1, Lp-cohomology is nondecreasing with p. Therefore, there is a critical p(M) such that H1,p(M) = 0 for p < p(M) and H1,p(M) = 0 for 6 p > p(M). It turns out that in the case of hyperbolic groups, this critical exponent can be interpreted as a kind of dimension of the ideal boundary, see [P2, G3, BP]. Indeed, it is always less than or equal to the infimal Hausdorff dimen- sion of metrics on theboundarycompatible with thenaturalquasiconformal structure, with equality in many examples, including lattices in rank one simple Lie groups and Fuchsian hyperbolic buildings. However, the inequal- ity may be strict, and this gives rise to examples of hyperbolic groups where the infimal Hausdorff dimension of metrics on the boundaryis not achieved, see [BP] and section 5. 8 3.4 Curvature pinching In higher degrees, torsion in Lp-cohomology sometimes sharply captures negative sectional curvature pinching. Let 1 δ < 0. Say a Riemannian manifold is δ-pinched if its sec- − ≤ tional curvature lies between a and δa for some a > 0. For example, real − hyperbolic space Hn is 1-pinched, complex hyperbolic space Hm, m 2, R C − ≥ quaternionic hyperbolic space Hm, m 2, and Cayley hyperbolic plane H2 H O ≥ are 1-pinched. −4 For real hyperbolic spaces Hn, torsion in Lp-cohomology vanishes most R of the time. In fact, in each degree, for at most one value of p, specificly, p = n 1 in degree k 2. This property extends to pinched Riemannian k−1 ≥ manifo−lds as follows. Theorem 3 [P6]. If Mn is simply connected and δ-pinched for some δ ∈ [ 1,0), then − n k p < 1+ − √ δ Tk,p(M) = 0. k 1 − ⇒ − This is sharp. Indeed, for every n 3, 2 k n 1 and δ [ 1,0), ≥ ≤ ≤ − ∈ − there exists ǫ > 0 and a δ-pinched homogeneous Riemannian manifold whose torsion does not vanish for p (1+ n k√ δ ǫ,1+ n k√ δ). ∈ k−1 − − k−1 − However, this comparison theorem− is not sharp fo−r negatively curved symmetricspaces. Forinstance,forcomplexhyperbolicspaceHm,fork = 2, C the pinching comparison theorem predicts that torsion vanishes for p < m, whereas it turns out that torsion still does not vanish for m p < 2m, see ≤ section 6. Inotherwords,ourapparentlyroughinvariant,torsioninLp-cohomology, not only detects subtle matters like sectional curvature pinching, but also distinguishes between homogeneous spaces which satisfy thesame curvature bounds. 4 Euler characteristic of negatively curved mani- folds 4.1 Hopf’s conjecture If M is a compact Riemannian manifold with constant sectional curvature 1, with even dimension n = 2m, then its Euler-Poincar´e characteristic is − 2 χ(M) = ( 1)m vol(M). − vol(S2m) 9 Indeed,theChern-WeilintegrandP (R)fortheEulerclassisahomogeneous χ polynomial of degree m in the curvature tensor R. The curvature tensors of the round sphere R and of real hyperbolic space R differ by a sign, 1 1 R = R . Therefore P (R ) = ( 1)mP (R ). Inte−grating this relates 1 1 χ 1 χ 1 − − − − Euler charateristics and volumes of constant curvature spaces. A similar calculation applies to other rank one locally symmetric man- ifolds M (the sphere is replaced by projective spaces over the complex, quaternion and octonion numbers). In all cases, ( 1)mχ(M) > 0. − Ifm = 1,theGauss-Bonnetformulaχ(M) = Rshowsthatχ(M) < 0 M as soon as R < 0. In higher dimension, it is notR so clear which negativity assumption on curvature should imply a sign for the Euler characteristic. A longstanding conjecture, attributed to H. Hopf, claims that the Euler- Poincar´e characteristic of a compact Riemannian 2m-manifold should be nonzero, andofthesamesignas( 1)m,ifitssectionalcurvatureisnegative. − 4.2 The Dodziuk-Singer conjecture It follows readily from the definition of L2-Betti numbers bi,2(M) that they can be used to compute Euler characteristic, n χ(M) = ( 1)ibi,2(M). − Xi=0 This has led J. Dodziuk and I. Singer to conjecture the following. Let M be an n-dimensional compact Riemannian manifold with negative sectional curvature. Let M˜ denote its universal covering. Then the reduced L2- cohomology Ri,2(M˜) vanishes if i = n/2, and does not vanish if i = n/2. 6 Note that the fact that M˜ admits a discrete cocompact isometric group action is essential, as shown by M. Anderson, [An]. Indeed, there exist simplyconnectedcompletenegativelycurvedmanifoldswhichadmitnonzero L2-harmonic forms in several degrees simultaneously. According to [P5] Th´eor`eme G,thereeven existRiemannian homogeneous spaces which admit nonzero L2-harmonic forms in all degrees 3, 4, ...,n 3 simultaneously. − 4.3 Gromov’s result Theorem 4 (M.Gromov, [G2]). Let M be a compact manifold with dimen- sion n = 2m. Assume that M admits a Ka¨hler metric. Assume that the fundamental group of M is Gromov-hyperbolic. Then ( 1)mχ(M) > 0. − In the rest of this section, we shall give a proof of this theorem. 10