Pinching estimates for negatively curved manifolds with nilpotent fundamental groups

a rX iv:mat h /45579v2[mat h.DG ]24Mar25PINCHING ESTIMATES FOR NEGATIVELY CUR VED MANIFOLDS WITH NILPOTENT FUNDAMENTAL GROUPS IGOR BELEGRADEK AND VITALI KAPOVITCH Abstract.Let M be a complete Riemannian metric of sectional curvature within [−a 2,−1]whose fundamental group contains a k -step nilpotent sub-group of finite index.We prove that a ≥k answering a question of M.Gro-mov.Furthermore,we show that for any ǫ>0,the manifold M admits a complete Riemannian metric of sectional curvature within [−(k +ǫ)2,−1].1.Introduction If the fundamental group of a complete pinched negatively curved manifold is amenable,it must be finitely generated and virtually nilpotent [BS87,Bow93,BGS85].In this paper we relate the nilpotency degree of the group to the pinching of the negatively curved metric.Theorem 1.1.Let M be complete Riemannian manifold with sectional cur-vature satisfying −a 2≤sec(M )≤−1.If Γis a k -step nilpotent subgroup of π1(M ),then a ≥k .In particular,if a ∈[1,2),then Γis abelian.If the cohomological dimension cd(Γ)of Γequals to dim(M )−1,which if dim(M )>2is equivalent to assuming that Γacts cocompactly on horospheres,Theorem 1.1follows from the proof of Gromov’s theorem of almost flat mani-folds (see [BK81,Corollary 1.5.2]),by combining the commutator estimate in almost flat horosphere quotients with the displacement estimate coming from the exponential convergence of geodesics.More recently,Gromov sketched in [Gro91,p.309]a proof of the more general estimatea ≥k2 ,where [x ]denotes the largest integer satisfying ≤x .If k ≤r +1,the estimate gives no information,so Gromov asked [Gro91,p.309]whether it can be im-proved to an estimate that is nontrivial for all cd(G )<dim(M ).Theorem 1.12IGOR BELEGRADEK AND VITALI KAPOVITCHprovides a satisfying answer that involves no dimension assumptions whatso-ever.The proof of Theorem1.1follows the original Gromov’s idea in[BK81],ex-cept that the commutator estimate is run in a“central”orbit of an N-structure given by the collapsing theory of J.Cheeger,K.Fukaya,and Gromov[CFG92]. In[BK]we proved the following classification theorem:Theorem1.2.[BK]A smooth manifold M with amenable fundamental group admits a complete metric of pinched negative curvature if and only if it is dif-feomorphic to the M¨o bius band,or to the product of a line and the total spaceaflat Euclidean vector bundle over a compact infranilmanifold.The”if”direction in Theorem1.2involves an explicit warped product construc-tion of a negatively pinched metric on the product of R and the total space of aflat Euclidean bundle over a closed infranilmanifold.By improving this warped product construction,we show that the pinching bounds provided by Theorem1.1are essentially optimal.Theorem1.3.If M be a pinched negatively curved manifold such thatπ1(M) has a k-step nilpotent subgroup offinite index,then M admits a complete Riemannian metric of sec(M)∈[−(k+ǫ)2,−1]for anyǫ>0.The metric constructed in Theorem1.3has cohomogeneity one,specifically M/Iso(M)is diffeomorphic to R(with the only exception when M is the M¨o bius band equipped with a hyperbolic metric).We do not know whether M in Theorem1.3always admits a complete metric with sec(M)∈[−k2,−1].This does happen for k=1,since as we show in[BK]any complete pinched negatively curved manifolds with virtually abelian fundamental group admits a complete hyperbolic metric.Another way to phrase the optimality of Theorem1.1is via the concept of pinching.Given a smooth manifold M,we define pinch diff(M)to be the infimum of a2≥1such that M admits a complete Riemannian metric of −a2≤sec(M)≤−1.If M admits no complete metric of pinched negative cur-vature,it is convenient to let pinch(M)diff=+∞.We then define pinch top(M) to be the infimum of all pinch diff(N)where N is homeomorphic to M,and define pinch hom(M)to be the infimum of pinch diff(N)’s where N is mani-fold with dim(N)=dim(M)that is homotopy equivalent to M.Of course, pinch diff(N)≥pinch top(M)≥pinch hom(N)≥1.In general,the pinching invariants are hard to estimate and even harder to compute(see[Gro91]and[Bel01,Section5]for surveys).Combining Theo-rems1.1–1.3,we compute the invariants in caseπ1(M)is virtually nilpotent. Corollary1.4.If M be a pinched negatively curved manifold such thatπ1(M) has a k-step nilpotent subgroup offinite index,then pinch diff(M)=pinch top(M)= pinch hom(M)=k2.PINCHING ESTIMATES FOR NILPOTENT GROUPS3 This work was partially supported by the NSF grants#DMS-0352576(Bele-gradek)and#DMS-0204187(Kapovitch).We are thankful to J.Cheeger for a discussion on collapsing.2.Proof of Theorem1.1A Riemannian metric is called A-regular if A={A i}is a sequence of nonnega-tive reals such that the norm of the curvature tensor satisfies||∇i R||≤A i.We call a metric regular if it is A-regular for some A.The collapsing theory works best for regular metrics,and the Ricciflow can be used to deform any metric with bounded sectional curvature to a complete Riemannian metric that is close to the original metric in uniform C1topology,has almost the same sectional curvature bounds,and is regular.(This fact has been known to some experts, but thefirst written account only recently appeared in[Kap]).Thus wefix an arbitraryδ>0and replace the given metric on M by a nearby A-regular metric g with sec g∈[−(a+δ)2,−1],and then prove that a+δ≥k,which would imply a≥k becauseδis arbitrary.Since the Riemannian covering of(M,g)corresponding toΓ≤π1(M)has the same curvature bounds as(M,g),we can assume thatπ1(M)=Γ.Denote the universal cover of M by X.If k=1,all we assert is a≥1which is trivially true,so we assume from now on that k>1.ThenΓfixes a unique point at infinity of the universal cover X of M(see e.g.[BS87]);let c(t)be a ray asymptotic to the point.Since sec(X)is bounded below,the family (X,c(t),Γ)has a subsequence(X,c(t i),Γ)that converges in the equivariant GH-topology topology to(X∞,c∞,Γ∞).Now sec(X)is also bounded above, the metric is regular,and X has infinite injectivity radius,hence the conver-gence(X,c(t i))→(X∞,c∞)is in fact in C∞topology.Then the quotients (X/Γ,p i)converge in pointed GH-topology to(X∞/Γ∞,p∞),where p i,p∞are the projections of c(t i),c∞,respectively.We now review the main results of[CFG92]as they apply to our situation;we refer to[CFG92]for terminology.Fixǫ,λwith0<ǫ≪1≪λ.By[CFG92, Theorems1.3,1.7,Proposition7.21],there are positive constantsρ,κ,ν,σ, depending only on n,ǫ,A such that for each large i,the manifold M carries an N-structure N i and an N i-invariant(ρ,κ)-round metric g i that isǫ-close to g in uniform Cλ-topology.Furthermore,there exists an orbit O i of N i such that(i)the metric on O i induced by g i has diam(O i)→0as i→∞,(ii)p i lies in theρ-neighborhood V i of O i,(iii)the normal injectivity radius of O i is≥ρ,(iv)the norm of the second fundamental form of O i is≤ν,and|sec(O i)|≤σ. (v)if˜V i→V i is the Riemannian universal cover,then˜V i admits a isometric4IGOR BELEGRADEK AND VITALI KAPOVITCHeffective action of a connected nilpotent Lie group G i that acts transitively on the preimage˜O i of O i under˜V i→V i,and intersectsπ1(V i)∼=π1(O i)in a normal subgroup that is cocompact in G i and has index≤κinπ1(V i).The above results are stated in[CFG92]in a different form,and their proofs are often omitted or merely sketched,so for reader’s convenience we briefly explain in the appendix how to deduce(i)-(iv).For(v)see[CFG92,pp.364–365]. Now we show that the inclusion V i→M isπ1-surjective for all large i.Indeed, let¯V i be a connected component of the preimage of V i under the cover X→M,and as before let˜V i,˜O i be the universal covers of V i,O i,respectively. Fix˜q i∈˜O i,and its projections,¯q i∈¯V i and q i∈O i⊂V i.By(i)-(ii)the sequence q i subconverges to some q∞∈X∞/Γ∞,hence for anyγ∈Γ,we have d(γ(¯q i),¯q i)→0as i→∞.So sinceΓisfinitely generated,if i is sufficiently large,then(i)-(ii)implies that¯V i contains the images of¯q i under somefinite generating set S ofΓ.By(iii)we see that¯V i contains the geodesic segment [¯q i,s(¯q i)]with s∈S,whose projection to V i⊂M represent the generator of π1(M,q i)∼=Γcorresponding to s.Hence the surjectionπ1(O i)→Γtakesπ1(O i)∩G i onto a normal subgroup of Γof index≤κ.The intersection of all normal subgroups ofΓof index≤κis a subgroupΓ0offinite index≤nκ2where n=dim M.(In fact,|Γ:Γ0|≤κ·νκwhereνκis the number of normal subgroups of index≤κ.SinceΓis nilpotent of cd(Γ)<n,it can be generated by<n elements,so there is a surjection from a rank n free group F n ontoΓ,andνκequals to the number of normal subgroups of F n of index≤κ,i.e.the number of elements in Hom(F n,Zκ), which is at most nκ.)Denote d(¯q i,γ(¯q i))by dγ.Below this notation is used for different distance functions,and each time we specify which metric we use.Since|Γ:Γ0|<∞,the nilpotency degree ofΓ0is k.Thus there areγj∈Γ0, j=1,...,k satisfying[γ1,[γ2,[γ3,[...[γk−1,γk]...]=γ=1.SinceΓ0lies in the image ofπ1(O i)→Γ,we can think of eachγj as acting on¯O i⊂X,where¯O i is the preimage of O i under the cover X→M.Note that one can chooseγj’s so that in the intrinsic metric on¯O i induced by g i ≤2nκ2·diam(O i).(Indeed,theΓ0-action on¯O i⊂X has a we have dγjfundamental domain F i of diameter≤nκ2·diam(O i).ThenΓ0is generated by S={s∈Γ0:s(¯F i)∩¯F i=∅},and each element of S has displacement at most2nκ2·diam(O i).Then there is a nontrivial k-fold commutator formed by elements of S,because otherwise the identity[a,bc]=[a,b]·[b,[a,c]]·[a,c] implies that any k-fold commutator inΓ0is trivial,so its nilpotency degree is< k).In particular,for the intrinsic metric induced on O i by g i the displacements →0as i→0.ofγj’s satisfy dγjPINCHING ESTIMATES FOR NILPOTENT GROUPS5 By(i)and(iv)we see that each O i with intrinsic metric induced by g i is almost flat,so the commutator estimate of[BK81,Proposition3.5(iii),Theorem2.4.1 (iii)]for the intrinsic metric on O i induced by g i gives(2.1)dγ≤c j dγj,where the constant c depends only on n,a.By Rauch comparison for Jacobifields,the normal exponential map is bi-Lipschitz on theρ-neighborhood in the normal bundle to O i,with Lipschitz constants depending on a,n,ρ.Hence the nearest point projection of theρ-tubular neighborhood of¯O i onto¯O i is K-Lipschitz for K=K(a,n,ρ),so any g i-geodesic of length≤2ρwith endpoints on¯O i is projected by the nearest point projection to a curve of length≤2ρK.Since the intrinsic displacements ofγj’s are<2ρfor all large i,the estimate(2.1)holds with a different c,for the distance function of the extrinsic metric g i,and again c only depends on n,a,ǫ,λ.Finally,since the distance functions of g and g i are bi-Lipschitz on B1(p i)), we get the same estimate(2.1)for the original metric g,with c depending on n,a,ǫ,k,λ.For the rest of the proof we work with displacements in metric g.Passing to asubsequence of p i’s,we canfind j such that dγj ≥dγlfor all l,i.Taking logswe getln dγ≤ln C+ln dγ1+...+ln dγk≤ln C+k ln dγjSince ln dγj <0and lim i→∞dγj=0,we deducelim supt→∞ln dγln dγj+k=kOn the other hand,by exponential convergence of geodesic rays,for any two elements ofΓ,and in particular forγ,γj we getlim supt→∞ln dγ6IGOR BELEGRADEK AND VITALI KAPOVITCH(b)an upper bound of the rotational part ofγj by a constant multiple of dγj that uses almostflatness[BK81,Proposition3.5(i)]).An alternative way to get(b)in our case is via the rotation homomorphismφ:Γ→O(n),introduced by B.Bowditch[Bow93],which is the holonomyof aΓ-invariantflat connection on X.A key property ofφis thatφ(γ)approximates the rotational part of anyγ∈Γwith error≤dγ.Now sinceany nilpotent subgroup of O(n)is abelian,φmust have a kernel of nilpotencedegree k−1.Hence,there is a(k−1)-fold commutator inΓwhose entrieslie in the kernel ofφ,and hence their rotational parts are bounded by their displacements.Repeating the argument at the end of the proof of Theorem1.1 for this commutator,we get a≥k−1.3.Infranilmanifolds are horosphere quotientsLet G be a simply-connected nilpotent Lie group acting on itself by left trans-lations,and let K be a compact subgroup of Aut(G),so that the semidirectproduct G⋊K acts on G by affine transformations.The quotient of G bya discrete torsion free subgroup of G⋊K is called an infranilmanifold.Weshowed in[BK]that any pinched negatively curved manifold with amenable fundamental group is either the M¨o bius band or product of an infranilmani-fold with R,and conversely,each of these manifolds admits an explicit warped product metrics of pinched negative curvature.This section contains a slight improvement of the warped product construction,that yields Theorem1.3.Consider the product of the above G⋊K-action onG with the trivial G⋊K-action on R.For the G⋊K-action on G×R,weprove the following.Theorem3.1.If G has nilpotence degree k,then for anyǫ>0,G×R admits a complete G⋊K-invariant Riemannian metric of sectional curvature within [−(k+ǫ)2,−1].Proof.The Lie algebra L(G)can be written asL(G)=L1⊃L2⊃···⊃L k⊃L k+1=0where L i+1=[L1,L i].Note that[L i,L j]⊂L i+j+1.Indeed,assume i≤jand argue by induction on i.The case i=1is obvious and the inductionstep follows from the Jacobi identity and the induction hypothesis,because[L i,L j]=[[L1,L i−1],L j]lies inspan([[L i−1,L j],L1],[[L1,L j],L i−1])⊂span([L i+j,L1],[L j+1,L i−1])=L i+j+1 The group K preserves each L i,so we can choose a K-invariant inner product , 0on L.LetF i={X∈L i: X,Y 0=0for Y∈L i+1}.PINCHING ESTIMATES FOR NILPOTENT GROUPS7 Then L=F1⊕···⊕F k.Define a new K-invariant inner product , r on L by X,Y r=h i(r)2 X,Y 0for X,Y∈F i,and X,Y r=0if X∈F i,Y∈F j for i=j,where h i are some positive functions defined below.This defines a G⋊K-invariant Riemannian metric g r on G.Letαi=i with i=1,···,k and a=k.Givenρ>0,we define the warping function h i to be a positive,smooth,strictly convex,decreasing function that is equal to e−αi r if r≥ρ,and is equal to e−ar if r≤−ρ;such a functionexists since a≥a i for each i.Thus h′i<0<h′′i,and the functions h′ih i areuniformly bounded away from0and∞.Define the warped product metric on G×R by g=s2g r+dr2,where s>0is a constant;clearly g is a complete G⋊K-invariant metric.A straightforward tedious computation(mostly done e.g.in[BW])yields for g-orthonormal vector fields Y s∈F s thatR g(Y i,Y j)Y j,Y i g=1h i h j,R g(Y i,Y j)Y l,Y m g=1∂r )∂h i, R g(Y i,∂∂r),Y j g=0if i=j,R g(∂2h j+h′l∂r,Y i)Y j,Y l g→0as s→∞,because| [Y j,Y i],Y l g|=s2| [Y j,Y i],Y l gr |≤s2Ck2|Y j|gr|Y i|gr|Y l|gr≤Ck2/s,where the last inequality holds since s|Y|gr=1for any g-unit vector Y.8IGOR BELEGRADEK AND VITALI KAPOVITCHIt follows that as s→∞,then R g uniformly converges to a tensor¯R whose nonzero components are¯R(Yi,Y j,Y j,Y i)=−h′i h′j∂r,∂h i.Thus g has pinched negative curvature for all large s.Finally,we show that for anyǫ>0there existsρsuch that sec g∈[−(k+ǫ)2,−1].Note thath′ih i=ln(h i)′′+(ln(h i)′)2.By construction|ln(h i)′|≤k.Also letρbe large enough,so that one can choose h i on[−ρ,ρ]to satisfy|ln(h i)′′|≪ǫ.Then for all sufficiently large s, the sectional curvature of g is within[−(k+ǫ)2,−1]. Proof of Theorem1.3.By[BK]if a pinched negatively curved manifold contains has a virtually k-step nilpotent fundamental group,then it is diffeomorphic to the quotient of G×R by a discrete torsion free subgroup of G⋊K.Thus we are done by Theorem3.1.Appendix A.On collapsing theoryThe purpose of this appendix is to outline the proof of the claims(i)-(iv)made in the proof of Theorem1.1.Some details can be found in[CFG92].Since g is regular,so is the corresponding metric˜g on the frame bundle F M. The balls(F B1(x),˜g)form an O(n)-GH-precompact family,where F B1(x)de-notes the frame bundle over the unit ball B1(x),x∈M.By[Fuk88]the closure of the family consists of regular Riemannian manifolds.So for an arbitrary se-quence p i∈M,the manifolds(F B1(p i),˜g)subconverge in O(n)-GH-topology to a pointed regular Riemannian manifold(Y,y).By the local version of Fukaya’sfibration theorem for some sequenceδi>0 satisfyingδi→0as i→∞,there exists for each large i an O(n)-equivariant δi-almost Riemannian submersion F B1(p i)→Y with nilmanifolds asfibers, which is also an O(n)-δi-Hausdorffapproximation.Furthermore,each F B1(p i) carries an O(n)-invariant N-structure˜N i whose orbits are the nilmanifoldfibers of the above submersion,and because of the O(n)-invariance,the structure de-scends to an N-structure N i on B1(p i).By[CFG92,Proposition7.21]F B1(p i) carries a metric˜g i that isǫ-close to˜g in Cλ-topology,and is both O(n)-invariant and˜N i-invariant.Hence˜g i induces unique Riemannian submersion metrics¯g i on Y,and g i on B1(x).To see(ii)-(iv),note that if l≤λ−2,then||∇l R¯gi||is bounded independently of i,so the sequence¯g i is precompact in Cλ−2-topology.Then by[PT99,PINCHING ESTIMATES FOR NILPOTENT GROUPS9 Lemma2.7]¯g i is precompact in O(n)-Cλ−2-topology,i.e.after pulling back by self-diffeomorphisms of Y,the metrics smoothly subconverge and share the same isometric O(n)-action.Thus there existsρ>0such that for each large i,the point y∈(Y,¯g i)lies in aρ-neighborhood of an O(n)-orbit that has normal injectivity radius≥ρ.The preimage O i of the O(n)-orbit under the Riemannian submersion(F B1(p i),˜g i)→(Y,¯g i)satisfies(ii)-(iii).Finally,(iii) implies the second fundamental form bound in(iv),which by Gauss formula gives a bound on|sec(O i)|.To see(i)note that the˜g-diameter of any orbit of˜N i is≤δi,so since˜g and ˜g i are bi-Lipschitz,the˜g i-diameter of any orbit of˜N i tends to zero as i→∞, and the same holds for orbits of N i because F B1(p i)→B1(p i)is distance non-increasing.Finally,the ambient diameter bound implies the intrinsic diameter bound,because Rauch comparison for Jacobifields gives bounds on bi-Lipschitz constants of the normal exponential map of O i,and in particular,the Lipschitz constant of the nearest point projection of theρ-tubular neighborhood of O i onto O i depends only on a,n,ρ,ǫ,λ.References[Bel01]I.Belegradek,Pinching,Pontrjagin classes,and negatively curved vector bundles, Invent.Math.144(2001),353–379.[BGS85]W.Ballmann,M.Gromov,and V.Schroeder,Manifolds of nonpositive curvature, Progress in Mathematics,vol.61,Birkh¨a user Boston Inc.,Boston,MA,1985.MR87h:53050[BK]I.Belegradek and V.Kapovitch,Classification of negatively pinched manifolds with amenable fundamental groups,preprint,/abs/math.DG/0402268. [BK81]P.Buser and H.Karcher,Gromov’s almostflat manifolds,Ast´e risque,vol.81,Soci´e t´e Math´e matique de France,1981.[Bow93] B.H.Bowditch,Discrete parabolic groups,J.Differential Geom.38(1993),no.3, 559–583.[BS87]M.Burger and V.Schroeder,Amenable groups and stabilizers of measures on the boundary of a Hadamard manifold,Math.Ann.276(1987),no.3,505–514. [BW]I.Belegradek and G.Wei,Metrics of positive Ricci curvature on bundles,to appear in Int.Math.Res.Not.,/abs/math.DG/0109167.[CE75]J.Cheeger and D.G.Ebin,Comparison theorems in Riemannian geometry,North-Holland Publishing Co.,Amsterdam,1975,North-Holland Mathematical Library,Vol.9.[CFG92]J.Cheeger,K.Fukaya,and M.Gromov,Nilpotent structures and invariant metrics on collapsed manifolds,J.Amer.Math.Soc.5(1992),no.2,327–372.[Fuk88]K.Fukaya,A boundary of the set of the Riemannian manifolds with bounded curva-tures and diameters,J.Differential Geom.28(1988),no.1,1–21.[Gro91]M.Gromov,Foliated Plateau problem.II.Harmonic maps of foliations,Geom.Funct.Anal.1(1991),no.3,253–320.[Kap]V.Kapovitch,Curvature bounds via Ricci smoothing,preprint, /abs/math.DG/0405569.10IGOR BELEGRADEK AND VITALI KAPOVITCH[PT99] A.Petrunin and W.Tuschmann,Diffeomorphismfiniteness,positive pinching,and second homotopy,Geom.Funct.Anal.9(1999),no.4,736–774.Igor Belegradek,School of Mathematics,Georgia Institute of Technology,At-lanta,GA30332-0160E-mail address:ib@Vitali Kapovitch,Department of Mathematics,University of California,Santa Barbara,CA93110E-mail address:vitali@。