Parameter rigid actions of simply connected nilpotent Lie groups
aa r X i v : . [ m a t h . G R ] M a y Parameter rigid actions of simply connectednilpotent Lie groups
Hirokazu Maruhashi ∗ Department of Mathematics, Kyoto University
Abstract
We show that for a locally free C ∞ -action of a connected and simplyconnected nilpotent Lie group on a compact manifold, if every real valuedcocycle is cohomologous to a constant cocycle, then the action is param-eter rigid. The converse is true if the action has a dense orbit. Usingthis, we construct parameter rigid actions of simply connected nilpotentLie groups whose Lie algebras admit rational structures with graduations.This generalizes the results of dos Santos [8] concerning the Heisenberggroups. Let G be a connected Lie group with Lie algebra g and M a C ∞ -manifoldwithout boundary. Let ρ : M × G → M be a C ∞ right action. We call ρ locallyfree if every isotropy subgroup of ρ is discrete in G . Assume that ρ is locally free.Then we have the orbit foliation F of ρ whose tangent bundle T F is naturallyisomorphic to a trivial bundle M × g .The action ρ is parameter rigid if any action ρ ′ of G on M with the same orbitfoliation F is C ∞ -conjugate to ρ , more precisely, there exist an automorphismΦ of G and a C ∞ -diffeomorphism F of M which preserves each leaf of F andhomotopic to identity through C ∞ -maps preserving each leaf of F such that F ( ρ ( x, g )) = ρ ′ ( F ( x ) , Φ( g ))for all x ∈ M and g ∈ G .Parameter rigidity of actions has been studied by several authors, for in-stance, Katok and Spatzier [3], Matsumoto and Mitsumatsu [4], Mieczkowski[5], dos Santos [8] and Ram´ırez [7]. Most of known examples of parameterrigid actions are those of abelian groups and nonabelian actions have not beenconsidered so much. ∗ Research Fellow of the Japan Society for the Promotion of Science G is contractible and M is compact. Let H be a Lie group. A C ∞ -map c : M × G → H is called a H -valued cocycle over ρ if c satisfies c ( x, gg ′ ) = c ( x, g ) c ( ρ ( x, g ) , g ′ )for all x ∈ M and g, g ′ ∈ G .A cocycle c is constant if c ( x, g ) is independent of x . A constant cocycle isjust a homomorphism G → H . H -valued cocycles c, c ′ are cohomologous if there exists a C ∞ -map P : M → H such that c ( x, g ) = P ( x ) − c ′ ( x, g ) P ( ρ ( x, g ))for all x ∈ M and g ∈ G .The action ρ is H -valued cocycle rigid if every H -valued cocycle over ρ iscohomologous to a constant cocycle. Proposition 1 ([4]) . If ρ is G -valued cocycle rigid, then it is parameter rigid.Remark. In [4] Matsumoto and Mitsumatsu assume that ρ has at least onetrivial isotropy subgroup, but this assumption is not necessary. Proposition 2 ([4]) . When G = R n , the following are equivalent:1. ρ is R -valued cocycle rigid.2. ρ is R n -valued cocycle rigid.3. ρ is parameter rigid.Remark. The equivalence of the first two conditions is obvious.In this paper we consider actions of simply connected nilpotent Lie groups.In [8], dos Santos proved that for actions of a Heisenberg group H n , R -valuedcocycle rigidity implies H n -valued cocycle rigidity and using this, he constructedparameter rigid actions of Heisenberg groups. To the best of my knowledge theseare the only known nontrivial parameter rigid actions of nonabelian nilpotentLie groups. We prove the following. Theorem 1.
Let N be a connected and simply connected nilpotent Lie group, M a compact manifold and ρ a locally free C ∞ -action of N on M . Then, thefollowing are equivalent:1. ρ is R -valued cocycle rigid.2. ρ is N -valued cocycle rigid.3. ρ is parameter rigid and every orbitwise constant real valued C ∞ -functionof ρ on M is constant on M . This theorem enables us to construct parameter rigid actions of nilpotentLie groups. The most interesting one is the following.2 heorem 2 ([7]) . Let N denote the group of all upper triangular real matriceswith on the diagonal, Γ a cocompact lattice of SL( n, R ) and ρ the action of N on Γ \ SL( n, R ) by right multiplication. If n ≥ , ρ is R -valued cocycle rigid.Remark. In [7], Ram´ırez proved more general theorems.
Corollary.
The above action ρ is parameter rigid. In Section 4 we construct parameter rigid actions of nilpotent groups usingTheorem 1. It is a generalization of dos Santos’ example. Let N be a connectedand simply connected nilpotent Lie group and Γ, Λ be lattices in N . Considerthe action of Λ on Γ \ N by right multiplication and let ˜ ρ be its suspended actionof N . Theorem 3. If Λ is Diophantine with respect to Γ , then the action ˜ ρ of N isparameter rigid. For the definition of Diophantine lattices, see Section 4.
Let G be a contractible Lie group with Lie algebra g , M a compact manifoldand ρ a locally free action of G on M with orbit foliation F . Let H be a Liegroup with Lie algebra h . We denote by Ω p ( F , h ) the set of all C ∞ -sections ofHom( V p T F , h ). The exterior derivative d F : Ω p ( F , h ) → Ω p +1 ( F , h )is defined since T F is integrable.By differentiating, H -valued cocycles over ρ are in one-to-one correspondencewith h -valued leafwise one forms ω ∈ Ω ( F , h ) such that d F ω + [ ω, ω ] = 0 . Proposition 3.
Let c , c be H -valued cocycles over ρ and let ω , ω be cor-responding differential forms. For a C ∞ -map P : M → H , the following areequivalent:1. c ( x, g ) = P ( x ) − c ( x, g ) P ( ρ ( x, g )) for all x ∈ M and g ∈ G .2. ω = Ad( P − ) ω + P ∗ θ where θ ∈ Ω ( H, h ) is the left Maurer-Cartanform on H . Corollary ([4]) . The following are equivalent:1. ρ is G -valued cocycle rigid.2. For each ω ∈ Ω ( F , g ) such that d F ω + [ ω, ω ] = 0 , there exist a endo-morphism Φ : g → g of Lie algebra and a C ∞ -map P : M → G suchthat ω = Ad( P − )Φ + P ∗ θ. ρ is givenby ω ∈ Ω ( F , R ) satisfying d F ω = 0. Two real valued cocycles ω , ω arecohomologous if and only if ω = ω + d F P for some C ∞ -function P : M → R .Leafwise cohomology H ∗ ( F ) of F is the cohomology of the cochain complex(Ω ∗ ( F , R ) , d F ). Thus H ( F ) is the set of all equivalence classes of real valuedcocycles.The identification T F ≃ M × g induces a map H ∗ ( g ) → H ∗ ( F ) where H ∗ ( g )is the cohomology of the Lie algebra g . By the compactness of M , this map isinjective on H ( g ). Hence we identify H ( g ) with its image. Note that H ( g )is the set of all equivalence classes of constant real valued cocycles. Thus realvalued cocycle rigidity is equivalent to H ( F ) = H ( g ).Notice that H ( F ) is the set of leafwise constant real valued C ∞ -functionsof F on M and H ( g ) consists of constant functions on M . Therefore theequivalence of 1 and 3 in Theorem 1 can be stated as follows: H ( F ) = H ( n )if and only if ρ is parameter rigid and H ( F ) = H ( n ). Let N be a simply connected nilpotent Lie group with Lie algebra n , M acompact manifold and ρ a locally free action of N on M with orbit foliation F .We first prove that N -valued cocycle rigidity implies real valued cocyclerigidity. There exist closed subgroups N ′ and A of N such that N ′ ⊳ N , N = N ′ ⋊ A and A ≃ R . Let c be any real valued cocycle over ρ . We regard c as a N -valued cocycle over ρ via the inclusion R ≃ A ֒ → N . By N -valued cocyclerigidity, there exist an endomorphism Φ of N and a C ∞ -map P : M → N such that c ( x, g ) = P ( x ) − Φ( g ) P ( ρ ( x, g )) for all x ∈ M and g ∈ N . Applyingthe natural projection π : N → A ≃ R , we obtain c ( x, g ) = ( π ◦ P )( x ) − ( π ◦ Φ)( g )( π ◦ P )( ρ ( x, g )). Thus c is cohomologous to a constant cocycle π ◦ Φ.Next we assume H ( F ) = H ( n ) and prove N -valued cocycle rigidity. Weneed the following two lemmata. Lemma 1.
Let V be a finite dimensional real vector space. Assume that ω ∈ Ω ( F , V ) satisfies the equation d F ω = ϕ , where ϕ ∈ Hom( V n , V ) isa constant leafwise two form. Then there exists a constant leafwise one form ψ ∈ Hom( n , V ) with ϕ = d F ψ .Proof. Since N is amenable, there exists a N -invariant Borel probability mea-sure µ on M . Define ψ ∈ Hom( n , V ) by ψ ( X ) = Z M ω ( X ) dµ where X ∈ n . Since ϕ ( X, Y ) = Xω ( Y ) − Y ω ( X ) − ω ([ X, Y ]) for all
X, Y ∈ n ,we obtain ϕ ( X, Y ) = − Z M ω ([ X, Y ]) dµ. d F ψ ( X, Y ) = − ψ ([ X, Y ]) = − Z M ω ([ X, Y ]) dµ = ϕ ( X, Y ) , hence d F ψ = ϕ .Set n = n , n i = [ n , n i − ]. Then n s = 0, n s +1 = 0 for some s . For each1 ≤ i ≤ s , choose a subspace V i with n i = V i ⊕ n i +1 , so that n = L si =1 V i . Lemma 2.
Let ω ∈ Ω ( F , n ) be such that d F ω + [ ω, ω ] = 0 . Decompose ω as ω = ξ + ω k + ω k +1 where ξ ∈ Ω ( F , L k − i =1 V i ) , ω k ∈ Ω ( F , V k ) and ω k +1 ∈ Ω ( F , n k +1 ) . If ξ is constant, then there exists ω ′ ∈ Ω ( F , n ) with d F ω ′ + [ ω ′ , ω ′ ] = 0 which iscohomologous to ω and such that ω ′ = ξ ′ + ω ′ k +1 where ξ ′ ∈ Ω ( F , L ki =1 V i ) is constant and ω ′ k +1 ∈ Ω( F , n k +1 ) .Proof. By cocycle equation,0 = d F ξ + d F ω k + d F ω k +1 + [ ξ, ξ ] + an element of Ω ( F , n k +1 ) . Comparing V k component, we see that d F ω k is constant. Hence by Lemma 1, d F ω k = d F ψ for some ψ ∈ Hom( n , V k ). Since we are assuming that H ( F ) = H ( n ), there exists ψ ′ ∈ Hom( n , V k ) and C ∞ -map h : M → V k such that ω k = ψ + ψ ′ + d F h. Put P = e h : M → N . Let x ∈ M and X ∈ T x F . Choose a path x ( t ) such that X = ddt x ( t ) (cid:12)(cid:12) t =0 . Let θ ∈ Ω ( N, n ) be the left Maurer-Cartan form on N . Then P ∗ θ ( X ) = ddt P ( x ) − P ( x ( t )) (cid:12)(cid:12)(cid:12) t =0 = ddt e − h ( x ) e h ( x ( t )) (cid:12)(cid:12)(cid:12) t =0 = ddt exp( − h ( x ) + h ( x ( t )) + an element of n k +1 ) (cid:12)(cid:12)(cid:12) t =0 = d F h ( X ) + an element of n k +1 . Thus P ∗ θ = d F h +an element of Ω ( F , n k +1 ). Note that Ad( P − ) = exp ad( − h )is identity on L ki =1 V i and preserves n k +1 . Hence ω − P ∗ θ = ξ + ψ + ψ ′ + an element of Ω ( F , n k +1 )= Ad( P − )( ξ + ψ + ψ ′ + an element of Ω ( F , n k +1 )) . ω be any N -valued cocycle. Using Lemma 2, we can exchange ω fora cohomologous cocycle whose V -component is constant. Applying Lemma 2repeatedly, we eventually get a constant cocycle cohomologous to ω . This proves N -valued cocycle rigidity.Next we assume that ρ is parameter rigid and H ( F ) = H ( n ). Let n i and V i be as above. Note that n s is central in n . Fix a nonzero Z ∈ n s .Let [ ω ] ∈ H ( F ). Let ω be the N -valued cocycle over ρ corresponding tothe constant cocycle id : N → N . We call ω the canonical -form of ρ . Fix a ǫ > η := ω + ǫωZ . η is an N -valued cocycle over ρ since d F η + [ η, η ] = d F ω + ǫ ( d F ω ) Z + [ ω , ω ] = 0 . Since M is compact, we can assume η x : T x F → n is bijective for all x ∈ M bychoosing ǫ > ρ ′ of N on M whose orbitfoliation is F and canonical 1-form is η . See [1]. By parameter rigidity ρ ′ isconjugate to ρ . Thus there exist a C ∞ -map P : M → N and an automorphismΦ of N satisfying ω + ǫωZ = Ad( P − )Φ ∗ ω + P ∗ θ. (1)Note that log : N → n is defined since N is simply connected and nilpotent. Letus decompose ω = P si =1 ω i , Φ ∗ ω = P si =1 ω ′ i and log P = P si =1 P i accordingto the decomposition n = L si =1 V i . Lemma 3.
Assume that P = · · · = P k − = 0 i.e. log P ∈ n k .1. If k < s , then there exist a C ∞ -map Q : M → N and an automorphism Ψ of N such that ω + ǫωZ = Ad( Q − )Ψ ∗ ω + Q ∗ θ and Q = · · · = Q k = 0 where log Q = P si =1 Q i .2. If k = s , then ω is cohomologous to a constant cocycle.Proof. For all X = ddt x ( t ) (cid:12)(cid:12) t =0 ∈ T x F , P ∗ θ ( X ) = ddt P ( x ) − P ( x ( t )) (cid:12)(cid:12)(cid:12) t =0 = ddt exp − s X i = k P i ( x ) ! exp s X i = k P i ( x ( t )) ! (cid:12)(cid:12)(cid:12) t =0 = ddt exp ( s X i = k ( P i ( x ( t )) − P i ( x )) + an element of n k +1 ) (cid:12)(cid:12)(cid:12) t =0 = ddt exp (cid:0) P k ( x ( t )) − P k ( x ) + an element of n k +1 (cid:1) (cid:12)(cid:12)(cid:12) t =0 = d F P k ( X ) + an element of n k +1 .
6e have Ad( P − )Φ ∗ ω = exp ad − s X i = k P i !! s X i =1 ω ′ i = s X i =1 ω ′ i + an element of n k +1 . Comparing the V k -component of (1) we get ω k + δ ks ǫωZ = ω ′ k + d F P k . When k = s the equation ωZ = ǫ − ( ω ′ s − ω s ) + d F ( ǫ − P s )shows that ω is cohomologous to a constant cocycle.If k < s , then d F P k = φ ◦ ω for some linear map φ : n → V k . Forany X ∈ n , let ˜ X denote the vector field on M determined by X via ρ . Wehave ˜ XP k = φ ( X ) and by integrating over an integral curve γ of ˜ X we get P k ( γ ( T )) − P k ( γ (0)) = φ ( X ) T for all T >
0. Since M is compact, φ ( X ) =0. Therefore d F P k = 0, so that P k is constant on each leaf of F . Thus P k is constant on M by our assumption. Put g := exp( − P k ) and Q := gP =exp (cid:0)P si = k +1 P i + an element of n k +1 (cid:1) . Then ω + ǫωZ = Ad( Q − g )Φ ∗ ω + ( L g − ◦ Q ) ∗ θ = Ad( Q − )Ψ ∗ ω + Q ∗ θ where Ψ ∗ := Ad( g )Φ ∗ .Applying Lemma 3 repeatedly, we see that ω is cohomologous to a constantcocycle.Finally we assume H ( F ) = H ( n ) and prove that ρ is parameter rigidand H ( F ) = H ( n ). Parameter rigidity of ρ follows from Proposition 1. Let f ∈ H ( F ). Fix a nonzero φ ∈ H ( n ) and denote the corresponding leafwise1-form on M by ˜ φ . Then f ˜ φ ∈ H ( F ) = H ( n ). Thus there exist ψ ∈ H ( n )and a C ∞ -function g : M → R such that f ˜ φ − ˜ ψ = d F g where ˜ ψ is the leafwise1-form corresponding to ψ . Choose X ∈ n satisfying φ ( X ) = 0. Let x ( t ) be anintegral curve of ˜ X where ˜ X is the vector field corresponding to X . We have f ( x ( t )) φ ( X ) − ψ ( X ) = ˜ X x ( t ) g = ddt g ( x ( t )). By integrating over [0 , T ], we get T ( f ( x (0)) φ ( X ) − ψ ( X )) = g ( x ( T )) − g ( x (0)). Since g is bounded, f ( x (0)) φ ( X ) − ψ ( X ) must be zero. Then f ( x (0)) = ψ ( X ) φ ( X ) and f is constant on M .This completes the proof of Theorem 1. Let us now construct real valued cocycle rigid actions of nilpotent groups. Forthe structure theory of nilpotent Lie groups, see [2]. A basis X , . . . , X n of n is7alled a strong Malcev basis if span R { X , . . . , X i } is an ideal of n for each i . IfΓ is a lattice in N , there exists a strong Malcev basis X , . . . , X n of n such thatΓ = e Z X · · · e Z X n . Such a basis is called a strong Malcev basis strongly based on Γ. Let Γ and Λ be lattices in N . Definition 1.
Λ is
Diophantine with respect to
Γ if there exists a strong Malcevbasis X , . . . , X n of n strongly based on Γ and a strong Malcev basis Y , . . . , Y n of n strongly based on Λ such that Y i = P ij =1 a ij X j for every 1 ≤ i ≤ n , where a ii is Diophantine.Let ρ be the action of Λ on Γ \ N by right multiplication. First we will provethe following. Theorem 4. If Λ is Diophantine with respect to Γ , then every real valued C ∞ cocycle c : Γ \ N × Λ → R over ρ is cohomologous to a constant cocycle. Note that X is in the center of n . Let π : N → ¯ N := e R X \ N be theprojection. Since Γ ∩ e R X = e Z X is a cocompact lattice in e R X , ¯Γ := π (Γ) = e R X \ Γ e R X is a cocompact lattice in ¯ N . Let ¯ n = R X \ n , then ¯ X , . . . , ¯ X n is astrong Malcev basis of ¯ n strongly based on ¯Γ.We will see that the naturally induced map ¯ π : Γ \ N → ¯Γ \ ¯ N is a principal S -bundle. Indeed, Γ \ Γ e R X ֒ → Γ \ N ։ Γ e R X \ N is a principal Γ \ Γ e R X -bundle and we haveΓ \ Γ e R X ≃ Γ ∩ e R X \ e R X = e Z X \ e R X ≃ Z \ R and e R X \ Γ e R X (cid:31) (cid:127) / / e R X \ N / / / / (cid:15) (cid:15) (cid:15) (cid:15) Γ e R X \ N ¯Γ \ ¯ N ∼ rrrrrrrrrr . Since Λ ∩ e R X = Λ ∩ e R Y = e Z Y is a cocompact lattice in e R X , ¯Λ := π (Λ)is a cocompact lattice in ¯ N . ¯ Y , . . . , ¯ Y n is a strong Malcev basis of ¯ n stronglybased on ¯Λ and ¯ Y i = P ij =2 a ij ¯ X j where a ii is Diophantine. Therefore ¯Λ isDiophantine with respect to ¯Γ.Since ¯ π is Λ-equivariant, the action ρ of Λ when restricted to e Z Y , preservesfibers of ¯ π .Let z ∈ ¯Γ \ ¯ N . Choose a point Γ x in ¯ π − ( z ). Then we have a trivialization ι Γ x : Z \ R ≃ ¯ π − ( z )of ¯ π − ( z ) given by ι Γ x ( s ) = Γ e sX x . Note that if we take another point Γ y ∈ ¯ π − ( z ), ι − y ◦ ι Γ x : Z \ R → Z \ R is a rotation.Let Y = aX where a is Diophantine. If we identify ¯ π − ( z ) with Z \ R by ι Γ x , then the action of e Y on Z \ R is s s + a .8et µ z be the normalized Haar measure naturally defined on ¯ π − ( z ), µ the N -invariant probability measure on Γ \ N and ν the ¯ N -invariant probabilitymeasure on ¯Γ \ ¯ N . For any f ∈ C (Γ \ N ), Z Γ \ N f dµ = Z ¯Γ \ ¯ N Z ¯ π − ( z ) f dµ z dν. (2) Lemma 4. ρ is ergodic with respect to µ .Proof. We use induction on n . If n = 1, ρ is an irrational rotation on Z \ R ,hence the result is well known. In general, Let f : Γ \ N → C be a Λ-invariant L -function with R Γ \ N f dµ = 0. Since the action of e Z Y on ¯ π − ( z ) is ergodic, f | ¯ π − ( z ) is constant µ z -almost everywhere. We denote this constant by g ( z ).Then g : ¯Γ \ ¯ N → C is ¯Λ-invariant measurable function. By induction, g isconstant ν -almost everywhere. By (2), this constant must be zero. Therefore f is zero µ -almost everywhere.Let c : Γ \ N × Λ → R be a C ∞ -cocycle over ρ . We must show that c is co-homologous to a constant cocycle c : Λ → R where c ( λ ) := R Γ \ N c ( x, λ ) dµ ( x ).Therefore we may assume that R Γ \ N c ( x, λ ) dµ ( x ) = 0 for all λ ∈ Λ, and we willshow that c is a coboundary. We prove this by induction on n . When n = 1, ρ is a Diophantine rotation on Z \ R , hence the result is well known. Lemma 5.
For all m ∈ Z , Z ¯ π − ( z ) c ( s, e mY ) dµ z ( s ) = 0 . Proof.
Fix m and put g ( z ) = R ¯ π − ( z ) c ( s, e mY ) dµ z ( s ). For any λ ∈ Λ, cocycleequation gives c ( x, λ ) + c ( xλ, e mY ) = c ( x, e mY ) + c ( xe mY , λ ). By integratingthis equation on ¯ π − ( z ), we get g ( zπ ( λ )) = g ( z ). Since the action of ¯Λ on ¯Γ \ ¯ N is ergodic, g is constant. By (2), g must be zero.Let f : Z \ R ι Γ x −−→ ¯ π − ( z ) c ( · ,e Y ) −−−−−→ R . We define h z ( ι Γ x ( s )) = X k ∈ Z \{ } ˆ f ( k ) − e πika e πiks . Then h z : ¯ π − ( z ) → R is C ∞ , since f is C ∞ and a is Diophantine. By Lemma5, we have c ( ι Γ x ( s ) , e Y ) = − h z ( ι Γ x ( s )) + h z ( ι Γ x e Y ) .
9f we choose another point Γ e s X x ∈ ¯ π − ( z ) to define h z , h z ( ι Γ x ( s )) = h z (Γ e sX x ) = h z ( ι Γ e s X x ( s − s ))= X k ∈ Z \{ } − e πika Z c (Γ e ( u + s ) X x, e Y ) e − πiku du e πik ( s − s ) = X k ∈ Z \{ } − e πika Z f ( u + s ) e − πiku du e πik ( s − s ) = X k ∈ Z \{ } ˆ f ( k ) − e πika e πiks , so that h z is determined only by z . Define h : Γ \ N → R by h | ¯ π − ( z ) = h z . Thenfor all x ∈ Γ \ N and m ∈ Z , c ( x, e mY ) = − h ( x ) + h ( xe mY ).Let U ⊂ ¯Γ \ ¯ N be open and σ : U → ¯ π − ( U ) a section of ¯ π . Then we havea trivialization Z \ R × U ≃ ¯ π − ( U ) which sends ( s, z ) to ι σ ( z ) ( s ) = Γ e sX σ ( z ).Hence h ( ι σ ( z ) ( s )) = X k ∈ Z \{ } − e πika Z c ( ι σ ( z ) ( u ) , e Y ) e − πiku du e πiks on ¯ π − ( U ). The following lemma shows h is C ∞ on Γ \ N . Lemma 6.
Let U ⊂ R n be open and let f : Z \ R × U → R be a C ∞ -function.Define h ( s, z ) = X k ∈ Z \{ } − e πika b f z ( k ) e πiks where f z ( u ) = f ( u, z ) . Then h : Z \ R × U → R is C ∞ .Proof. Let V be open such that ¯ V ⊂ U and ¯ V is compact. We will show that h is C ∞ on Z \ R × V . Choose constants C, α > |− e πika | ≥ C | k | − α for all k ∈ Z \ { } .We will first prove that h is continuous. Since for any m ∈ Z > , ∂ m f z ∂s m ( s ) = X k ∈ Z (2 πik ) m b f z ( k ) e πiks in L ( Z \ R ), (cid:13)(cid:13)(cid:13) ∂ m f z ∂s m (cid:13)(cid:13)(cid:13) = X k ∈ Z | (2 πik ) m b f z ( k ) | ≥ (2 π ) m | k | m | b f z ( k ) | ≥ | k | m | b f z ( k ) | . Since (cid:13)(cid:13)(cid:13) ∂ m f z ∂s m (cid:13)(cid:13)(cid:13) = (cid:18)R (cid:12)(cid:12)(cid:12) ∂ m ∂s m f ( s, z ) (cid:12)(cid:12)(cid:12) ds (cid:19) is continuous in z , there exists M > (cid:13)(cid:13)(cid:13) ∂ m f z ∂s m (cid:13)(cid:13)(cid:13) < M for every z ∈ ¯ V . Hence for all k ∈ Z and z ∈ ¯ V ,10 k | m | b f z ( k ) | ≤ M . Therefore, for any z ∈ ¯ V , X k ∈ Z \{ } (cid:12)(cid:12)(cid:12) − e πika b f z ( k ) e πiks (cid:12)(cid:12)(cid:12) ≤ C − X k ∈ Z \{ } | k | | k | α +2 | b f z ( k ) |≤ C − M X k ∈ Z \{ } | k | < ∞ . This implies continuity of h on Z \ R × ¯ V .We have ∂h∂s ( s, z ) = X k ∈ Z \{ } πik − e πika b f z ( k ) e πiks . Thus a similar argument shows that ∂h∂s is continuous.Let z = ( z , . . . , z n ). For any z ∈ ¯ V , (cid:12)(cid:12)(cid:12) ∂∂z j (cid:18) − e πika b f z ( k ) e πiks (cid:19)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) − e πika \ ∂f∂z j ( · , z )( k ) e πiks (cid:12)(cid:12)(cid:12) ≤ C − | k | | k | α +2 (cid:12)(cid:12)(cid:12) \ ∂f∂z j ( · , z )( k ) (cid:12)(cid:12)(cid:12) ≤ C − M ′ | k | ∈ L ( Z \ { } ) . Thus ∂h∂z j ( s, z ) = X k ∈ Z \{ } − e πika \ ∂f∂z j ( · , z )( k ) e πiks . Hence ∂h∂z j is continuous by an argument similar to those above. For higherderivatives of h , continue this procedure.Set c ( x, λ ) = c ( x, λ ) + h ( x ) − h ( xλ ). c : Γ \ N × Λ → R is a C ∞ -cocycleand c ( x, e mY ) = 0. Thus for any λ ∈ Λ, cocycle equation implies c ( x, λ ) = c ( xe Y , λ ). Since the action of e Z Y on ¯ π − ( z ) is ergodic, c ( x, λ ) is constant on¯ π − ( z ). Therefore we can define a cocycle ¯ c : ¯Γ \ ¯ N × ¯Λ → R by ¯ c (¯ π ( x ) , π ( λ )) = c ( x, λ ). Indeed, if ¯ π ( x ) = ¯ π ( y ) and π ( λ ) = π ( λ ′ ), then there exists a m ∈ Z with λ = e mY λ ′ , so that c ( x, λ ) = c ( x, e mY λ ′ ) = c ( xe mY , λ ′ ) = c ( y, λ ′ ) . Furthermore, Z ¯Γ \ ¯ N ¯ c ( x, π ( λ )) dν ( z ) = Z ¯Γ \ ¯ N Z ¯ π − ( z ) c ( s, λ ) dµ z ( s ) dν ( z )= Z Γ \ N c ( x, λ ) dµ ( x ) = 0 .
11y induction, there exists a C ∞ -function P : ¯Γ \ ¯ N → R such that ¯ c ( z, π ( λ )) = − P ( z ) + P ( zπ ( λ )). Put Q = P ◦ ¯ π . Then c ( x, λ ) = ¯ c (¯ π ( x ) , π ( λ )) = − Q ( x ) + Q ( xλ ). This proves Theorem 4.Let ˜ ρ : M × N → M be the suspension of ρ : Γ \ N × Λ → Γ \ N where M = Γ \ N × Λ N is a compact manifold. Then ˜ ρ is locally free and let F be itsorbit foliation. We have H ( F ) ≃ H (Λ , C ∞ (Γ \ N ))by [6] where the right hand side is the first cohomology of Λ-module C ∞ (Γ \ N )obtained by ρ . It is easy to prove that Hom(Λ , R ) → H (Λ , C ∞ (Γ \ N )) isinjective. By Theorem 4, H (Λ , C ∞ (Γ \ N )) = Hom(Λ , R ) . Lemma 7. dim Hom(Λ , R ) = dim H ( n ) . Proof.
Recall that [
N, N ] \ Λ[ N, N ] is a cocompact lattice in [
N, N ] \ N and that[Λ , Λ] \ (Λ ∩ [ N, N ]) is finite. Since0 → [Λ , Λ] \ (Λ ∩ [ N, N ]) → [Λ , Λ] \ Λ → [ N, N ] \ Λ[ N, N ] → , Λ] \ Λ = rank[
N, N ] \ Λ[ N, N ] = dim[
N, N ] \ N. Thus dim Hom(Λ , R ) = dim Hom([Λ , Λ] \ Λ , R )= rank[Λ , Λ] \ Λ= dim[
N, N ] \ N = dim Hom R ([ n , n ] \ n , R )= dim H ( n ) . Therefore we obtain H ( F ) = H ( n ) . This proves Theorem 3.
Let n Q be a rational structure of n . We construct Diophantine lattices when n Q admits a graduation. Namely, we assume that n Q has a sequence V i of Q -subspaces such that n Q = L ki =1 V i and [ V i , V j ] ⊂ V i + j . Let X , . . . , X n be a Q -basis of n Q such that X , . . . , X i ∈ V k , X i +1 , . . . , X i ∈ V k − , . . . ,12 i k − +1 , . . . , X n ∈ V . Then X , . . . , X n is a strong Malcev basis of n withrational structure constants. Multiplying X , . . . , X n by a integer if necessary,we may assume that Γ := e Z X · · · e Z X n is a cocompact lattice in N . Let α be aroot of a irreducible polynomial of degree k + 1 over Q . Since α, α , . . . , α k areirrational algebraic numbers, they are Diophantine. If we define a linear map ϕ : n → n by ϕ ( X ) = α i X for X ∈ V i ⊗ R , then ϕ is an automorphism of Liealgebra n . Put Y i = ϕ ( X i ). Y , . . . , Y n is a strong Malcev basis of n stronglybased on Λ := e Z Y · · · e Z Y n . Thus Λ is Diophantine with respect to Γ. Acknowledgement
The author would like to thank his advisor, Masayuki Asaoka, for helpful com-ments.
References [1] M. Asaoka.
Deformation of lacally free actions and the leafwise cohomology .arXiv:1012.2946.[2] L. Corwin and F. P. Greenleaf.
Representations of nilpotent Lie groups andtheir applications. Part 1:Basic theory and examples . Cambridge studies inadvanced mathematics, vol. 18, Cambridge University Press, Cambridge,1990.[3] A. Katok and R. J. Spatzier.
First cohomology of Anosov actions of higherrank abelian groups and applications to rigidity . Inst. Hautes ´Etudes Sci.Publ. Math. (1994), 131–156.[4] S. Matsumoto and Y. Mitsumatsu. Leafwise cohomology and rigidity ofcertain Lie group actions . Ergod. Th. & Dynam. Sys. (2003), 1839–1866.[5] D. Mieczkowski. The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory . J. Mod. Dyn. (2007), 61–92.[6] M. S. Pereira and N. M. dos Santos. On the cohomology of foliated bundles .Proyecciones (2)(2002), 175–197.[7] F. A. Ram´ırez. Cocycles over higher-rank abelian actions on quotients ofsemisimple Lie groups . J. Mod. Dyn. (2009), 335–357.[8] N. M. dos Santos. Parameter rigid actions of the Heisenberg groups . Ergod.Th. & Dynam. Sys.27