The first Cohomology of Affine Z

The first Cohomology of Affine Z

-actions on Tori and applications to rigidity*

Richard Urzúa Luz

Abstract. Let ϕ a minimalaffineZ^p-action on the torus T^q, p ≥ 2 andq ≥ 1.

The cohomology ofϕ (see definition below) depends on both the algebraic properties of the induced action onH¹(T^q,Z)and the arithmetical properties of the translation cocycle. We give a Diophantine condition that characterizes those affine actions whose first cohomology group is finite dimensional. In this case it is necessarily isomorphic to R^p.Thus theR^p-actionFϕobtained by suspension ofϕisparameter rigid, i.e., any other R^p-action with the same orbit foliation is smoothly conjugate to a reparametrization of F_ϕby an automorphism ofR^p.

Keywords: minimal action, cocycle rigidity, diophantine condition, parameter rigidity, cohomology of ergodic actions.

Mathematical subject classification: 37B05, 57S25, 37C85; 58E40, 11K60, 53C24, 20J06.

1 Introduction

Let Affine(T^q)denote the group of affine transformations of the torus T^q.By an affineZ^p-action onT^qwe mean a homomorphismϕ ofZ^p into Affine(T^q).

The actionϕinduces aZ^p-action by automorphisms of the ringC^∞(T^q)of the smooth functions onT^qgiven by·f =f ◦ϕ()for∈Z^pandf ∈C^∞(T^q) defining aZ^p-module structureC_ϕ^∞(T^q) onC^∞(T^q). Thecohomology of the actionϕ is by definition the cohomologyH^∗(Z^p, C_ϕ^∞(T^q))of theZ^p- module C_ϕ^∞(T^q),see for example [2].

The investigation of the cohomology of ergodic actions of higher rank abelian groups(e.g.,Z^k andR^k fork ≥ 2) has attracted considerable interest in recent years due to its connection with cocycle rigidity phenomena, i.e. every smooth

Received 17 January 2003.

*Partially supported by CNPq fellowship by Fondecyt Grant 1000047 and DGICT-UCN and fundación Andes, Chile.

cocycle is cohomologous to a constant one. Some cocycle rigidity phenomena appear in connection with hyperbolicity. For example A. Katok and R. Spatzier [10], [11] and [12] showed that real cocycles of certain Anosov actions ofZ^p orR^p(standard Anosov) are cohomologous to constant cocycle. Similar results are obtained by A. Katok and K. Schmidt [9] for mixing expansive actions ofZ^p on the automorphism group of an abelian group.

Another source of cocycle rigidity are certain Z^p-actions on the group of translations of the torusT^q. This actions are determined by linear foliations with minimal leaves transverse to the fibers of the canonical projection ofT^p+q to T^p, see [1], [22], [15], [20], [6], [4].

In this paper we give aDiophantine conditionfor affine Z^p-actions on tori.

Our main result states that the first cohomology group of an affine action is finite dimensional if and only if the action satisfies the Diophantine condition, see Section 1.2 for the precise statement of the result. This extends the work of J.L.

Arraut and N.M. dos Santos done for actions of translations.

A consequence of our result is that, if the first cohomology group of the action is finite dimensional, then it is necessarily isomorphic toR^p.

1.1 Geometrical consequences

Our main result, characterizing those affine actions with finite dimensional first cohomology group, has the following geometrical consequence.

Associated to eachZ^p- action onT^q we have its suspension foliated bundle T^q→(M,F)→^τ T^qand anR^p-action by automorphisms of the projectionτ, called thecanonical action ofτ.The leafwise cohomologyH^∗(F)is isomorphic to the cohomology ofϕ[22].

We say that a locally-freeR^p-actionis parameter rigid if any otherR^p-action with the same orbit foliation is smooth conjugate to it, up to a reparametrization by an automorphism ofR^p. S. Matsumoto and Y. Mitsumatsu [19] proved that a locally-free action ofR^p is parameter rigid if and only if the first cohomol- ogy group is isomorphic toR^p. Thus the locally freeR^p-action induced by an affineZ^p-action onT^q is parameter rigid if and only if the action satisfies the Diophantine condition, as defined below (Section 1.2).

This extends what was done by J.L. Arraut, N.M. dos Santos [1] and J. Moser [20] for action of translations. They gave a definition of a Diophantine Z^p- action by translations onT^q and proved that the cohomology of such an action is isomorphic to the cohomology ofT^qwith real coefficients.

1.2 Statements of Results

An affineZ^p-actionϕon the torusT^qinduces aZ^p-action by automorphisms of H1(T^q,Z) ∼= Z^q andH¹(T^q,Z), denoted by ϕ∗ andϕ^∗, respectively. The set σ (ϕ∗)of all eigenvalues of allϕ∗()is referred to as thespectrum ofϕ∗. Ifϕis minimal thenσ (ϕ∗)= {1}. The cohomology ofϕdepends on both the algebraic properties ofϕ^∗and the arithmetical properties of the translation cocycle. The isotropy group ofϕ^∗atk∈Z^qwill be denoted byI (k). Each isotropy groupIof ϕ^∗is a direct summand, i.e., there is a subgroupKofZ^psuch thatZ^p=I⊕K. An affineZ^p-actionϕonT^qsatisfies the irrationality condition if fork∈Z^q such that dimI (k)≥p−1 there is∈I (k)so that

k, α() ∈Z

whereϕ()˜ =ϕ∗()+α()is a lifting ofϕ()to the coveringR^qand , is the usual inner product onR^q. Letbe the set of fixed points ofϕ^∗i.e.

= {k∈Z^q|ϕ^∗()k =kfor all∈Z^p}.

We note that ifσ (ϕ∗)= {1}then=0.An affineZ^p-actionϕonT^qsatisfies a Diophantine condition on if there are constantsβ >0 andC >0 so that for eachk∈− {0}there isj, 1≤j ≤psatisfying

k, α(e_j) ≥C|k|^−β

where{e1, . . . , ep} is the canonical basis of Z^p, |x| = sup_j|xj| andx = inf{|x−l| |l∈Zⁿ}.

Our main result is

Theorem 1. Letϕbe an affineZ^p-action onT^qwithσ (ϕ∗)= {1},wherep≥2 andq ≥1.Suppose thatϕsatisfies the irrationality condition. Then the following statements are equivalent

1. ϕ satisfies a Diophantine condition on. 2. H¹(Z^p, C_ϕ^∞(T^q))∼=R^p.

3. H¹(Z^p, C_ϕ^∞(T^q))is Hausdorff.

The main feature of an action by translations is that the induced action on cohomology is trivial and this greatly simplifies the calculation of the cohomol- ogy. This is not true for a general affine action and in fact the induced action on cohomology can be rather complicated. For this reason Theorem 1 is a nontrivial generalization of the corresponding result in [1].

The basic conjecture is that a minimal smoothZ^p-action onT^qwhose first co- homology is isomorphic toR^pshould be smoothly conjugate to its corresponding affineZ^p-action. This conjecture is supported by [16] and [26].

1.3 Examples

We now give the simplest example of affine Z²-action which does not act by translations and the first cohomology group is isomorphic toR².

Letϕ be an affineZ²-action onT²generated on the coveringR²by

˜

ϕ1(x, y)=(x+α, x+y) and

ϕ˜²(x, y)=(x, y+β).

The actionϕsatisfies the irrationality condition and Diophantine condition on if and only ifαis a Diophantine number andβ is irrational number. The second cohomology depends onβ. It is isomorphic toRifβis Diophantine and it is non- Hausdorff ifβis Liouville [15]. Contrasting with this aZ^p-action by translations onT^q is Diophantine if only if the first cohomology is finite dimensional and thus its cohomology is necessarily isomorphic to the real cohomology ofT^p[1].

An other interesting example is that of an affineZ²-action on torusT⁵such that the induced action on cohomology has infinitely many isotropy groups and the first cohomology group of the action is finite dimensional. The action is generated on the coveringR⁵by

ϕ˜¹(x¹, x², x³, x⁴, x⁵)=(x¹+x⁴, x²+x⁵, x³+β, x⁴+α, x⁵) and

ϕ˜2(x1, x2, x3, x4, x5)=(x1+γ , x2+x4, x3+x5, x4, x5+α).

Ifαis a Diophantine number andα, β andγ are linearly independent over the rational numbers then the action satisfies the hypothesis of the Theorem 1.

2 On affine minimalZ^p-actions onT^q

Letϕ be aZ^p-action onT^q.Sinceϕ∗acts by automorphisms ofZ^q it induces aZ^p- action ϕ⁰ by automorphisms of T^q referred to as thelinear part of the actionϕ. The actionϕ can be written uniquely asϕ() = τ ()ϕ0(), ∈ Z^p whereτ :Z^p−→T^qis a cocycle overϕ0i.e.τ (+)=τ ()ϕ0()τ ()for all , ∈Z^p. We callτ is the translation cocycle ofϕ. A liftingα :Z^p −→R^q, on the coveringR^qπ◦α=τ ofτ is not in general a cocycle over the actionϕ∗

butα(+)= α()+ϕ∗()α()+k(, ),k(, )∈ Z^q. The actionϕ can be lift to aZ^p-actionϕ˜ =ϕ∗+αif and only if the affineZ^p-actionϕis isotopic to its linear partϕ0.

Proposition 2.1. If an affineZ^p-actionϕonT^q is minimal thenσ (ϕ∗)= {1}.

Proof. We first show that 1∈σ (ϕ∗())for all∈Z^p. In fact, if 1∈σ (ϕ∗(0) for some⁰ ∈ Z^p then ϕ(l⁰)has a fixed point x⁰ = π(x˜⁰) wherex˜⁰ = (id − ϕ∗(0))⁻¹α(0),α :Z^p −→ R^q being any lifting of the translation cocycle of ϕandπ :R^q −→T^qbe the covering map. So the closure of theϕ-orbit ofx0

is left fixed byϕ(⁰)and asϕis minimal thenϕ(⁰)is the identity map ofT^q, giving a contradiction.

Assume now there exist0 ∈ Z^p such that σ (ϕ∗(0)) = {1}. Letp(x) = (x−1)^mq(x)be the primary decomposition of the minimal polynomial ofϕ∗(⁰) overQ[x].

Thus there is decompositionR^q =E1⊕E2, whereE1=ker(id −ϕ∗(0))^m andE² = kerq(ϕ∗(⁰)) are invariant by the actionϕ∗, andE¹ has a basis in Z^p. Decompose a lifting of the translation cocycle of ϕ asα() = α1()+ α2(), ∈ Z^p. Now it is easy to verify that the translationT˜ζ = id+ζ, ζ = (id −ϕ∗(0))⁻¹α1(0) conjugate ϕ with the action ϕ1 given on the covering R^qbyϕ˜¹()=ϕ∗()+α¹()+r(), r()∈ E²∩Q^q, r(⁰)=0,i.e. ϕ¹()= T_σ⁻¹◦ϕ()◦Tσ, ∈ Z^p andϕ1is not minimal since the orbit ofπ(0)lies in finitely many tori of dimension less thanqif degree ofq(x) >1.

Forp=1 the above proposition is a result of F.J. Hahn [5].

There is an algebraic characterization of minimal actions.

Let⊂Z^q be the set of fixed points ofϕ^∗, i.e.,

=

k∈Z^q |ϕ^∗()k =k, for all∈Z^p .

We say thatϕsatisfies the irrationality condition onif for eachk∈ − {0}

there exist∈Z^psuch that

k, α() ∈Z

whereϕ()˜ =ϕ∗()+α()is any lift ofϕ()to the coveringR^q. We prove

Theorem 2.Letϕbe an affineZ^p-action onT^qwithσ (ϕ∗)= {1}.The following data are equivalent

1. ϕ satisfies the irrationality condition on.

2. ϕ is ergodic with respect to the Haar measure onT^q. 3. ϕ is uniquely ergodic.

4. ϕ has a dense orbit.

5. ϕ is a minimal action.

Proof. 1. ⇒ 2. It is sufficient to show that any functionf ∈L²(T^q, m)such that

f ◦ϕ()=f a.e.

for all∈Z^pis constant a.e. In fact, the Fourier coefficients off satisfy f◦ϕ()(k)=e^{2πi k,(ϕ∗())−1α()}f(ϕ^∗())⁻¹k =f (k) (1) Ifk∈−{0}, then from(1)we obtain thatfk =e^{2πi k,α()}fkande^{2πi k,α()}=1 sinceϕsatisfies the irrationality condition on. Thusfk =0, for allk∈−{0}.

Ifk∈− {0}there exist0∈Z^p such thatϕ^∗(0)⁻¹k=k.Asσ (ϕ^∗(0))= {1}, the sequence{ϕ^∗(n0)k}is infinite. For eachn ∈ Zwe take = n0 in (1),as lim_n→±∞|ϕ^∗(n⁰)k| = lim_n→±∞|ϕ^∗(⁰)ⁿk| = ∞. By The Riemann - Lebesgue Theorem givesfk =0.

2.⇒4. It follows from the fact thatϕis ergodic with respect to the Haar measure onT^q.

4.⇒ 5. Ifq = 1, a dense orbit trivially implies minimality becauseϕ acts by rotations of the circle. Let us assume that the result is true forq =n−1. We show it is true forq =n. Sinceσ (ϕ∗)= {1},we may assume that the matrices ϕ∗()are lower triangular and thus have

ϕ()(z¹, z²)=(ϕ¹()(z¹), β()z²B()(z¹)) (2) wherez1 ∈ Tⁿ⁻¹,z2 ∈T¹,β()∈ T¹for all∈ Z^p,ϕ1is an action ofZ^p on the affine group ofTⁿ⁻¹, andB()is an homomorphism ofTⁿ⁻¹inT¹. Observe that if the ϕ-orbit of (z⁰1, z⁰2) by ϕ is dense in Tⁿ, then the ϕ1-orbit of z⁰1 is dense inTⁿ⁻¹, so by the induction assumptionϕ1is minimal. Therefore for each z=(z¹, z²)∈Tⁿ⁻¹×T¹we have

(z)= {ϕ()(z)|∈Z^p} ∩({z⁰1} ×T¹)=φ.

As the orbit ofϕat every point of{z1⁰} ×T¹is dense inTⁿ,then(z)=Tⁿ. 3.⇒ 5. This follows from the fact that the only invariant measure is the Haar measure.

2.⇒3. Ifq =1, thenϕacts as rotations of the circle, thenϕis uniquely ergodic.

Let us assume that it is true forq =n−1. We show that it is true forq =n. From(2)we see that ifϕ is ergodic thenϕ1is ergodic and by the induction hypothesisϕ1is uniquely ergodic,thereforeϕis uniquely ergodic [21].

5.⇒1. Ifϕdoes not satisfies the irrationality condition on,then there existk0

so that k0, α() ∈Zfor all∈Z^pand the function cos(2π k0, x)is invariant

byϕthereforeϕis not minimal.

Problem 1. Let Aa homomorphism of Z^p into GL(q,Z) with spectrum 1.

Give a necessary and sufficient condition for the existence of a minimal affine Z^p -action onT^q whose induced action on the first homology group ofT^q is preciselyA.

3 The first cohomology group of affineZ^p-actions on tori

A functionf :Z^p −→C^∞(T^q)is a 1-cocycleof an affineZ^p- actionϕonT^qif f (+)=f ()+f ()◦ϕ(),for allandinZ^p. The setZ¹(Z^p, C_ϕ^∞(T^q))of all 1-cocyclesf :Z^p −→C^∞(T^q)of an affineZ^p-actionϕonT^qis an abelian group. A 1-cocycle f is trivial or a 1-coboundary if f () = h−h◦ϕ() for some h ∈ C^∞(T^q) and all ∈ Z^p. The set B¹(Z^p, C_ϕ^∞(T^q)) of all 1- coboundaries ofϕ is a subgroup ofZ¹(Z^p, C_ϕ^∞(T^q))andH¹(Z^p, C_ϕ^∞(T^q)) = Z¹(Z^p, C^∞(T^q))/B¹(Z^p, C^∞(T^q))is the first cohomology group of ϕ.

Since the Haar measuremis invariant byϕthen to each 1-cocyclefofϕthere corresponds a homomorphismm(f ) : Z^p −→ R, m(f )() =

T^qf ()dm, ∈Z^p. Thus we have the decomposition

Z¹(Z^p, C_ϕ^∞(T^q))=H om(Z^p,R)⊕kerm (3) where kerm= {f ∈Z¹(Z^p, C_ϕ^∞(T^q))|m(f )()=0 for all∈Z^p}.

We now prove the Theorem 1.

Proof. The implication 2⇒3 is trivial. The main implication 1⇒2 is proved in Section 5; we now prove the implication 3⇒1.

Suppose thatϕdoes not verify any Diophantine condition on the fixed points ofϕ^∗. We show that the first cohomology group ofϕ is non-Hausdorff. In fact there existj, 1 ≤ j ≤ p and a sequence{k_s}_s≥1 insuch that|k_s| → ∞as s→ ∞

0< ks, α(ej) = max

1≤i≤p ks, α(ei)<|ks|^−s. (4) We construct a non-trivial cocycle sequencef :Z^p−→C^∞(T^q)ofϕwhere as beforeϕ()˜ =ϕ∗()+α()is a lifting ofϕ()to the coveringR^q which is the limit in theC^∞topology of coboundaries. For each∈ Z^p− {0}we consider the functionf ()given by the Fourier series

f ()=^∞

s=1

1−e^{−2πi ks,α()} φ−ks +

1−e^2πi<ks,α() φks

where as usualφk =e^{2πi k,x}.

From(4)we get

0<|| ks, α()|| ≤ || · || ks, α(ej)|| ≤ || · |ks|^−s (5) which shows that the above Fourier series converges in theC^∞topology to a C^∞functionf ():T^q −→R.

From(5)we see thatf :Z^p −→C^∞(T^q), → f ()is a cocycle since the sequence{ks}consists of fixed points of the actionϕ^∗. The partial sumsSn()of the Fourier series off ()also define cocyclesSn():Z^p −→ C^∞(T^q)which are clearly coboundaries. Now we show thatf is not a coboundary. Suppose thatf ()=h−h◦ϕ()for someC^∞functionhand all∈Z^p.

Thus fors ≥1 we have

f ()(k s)=(1−e^{2πi ks,α()})h(ks)

andh(ks)=1,contradicting the Riemann-Lebesgue theorem.

The distributions onT^qwhich are invariant byϕplay an important role on the computation of the cohomology ofϕ,and are discussed in the next section.

4 Invariant distributions and the first cohomology group

Letϕ be an affineZ^p- action onT^q andI be an isotropy group of the induced actionϕ^∗. Choose a subgroupH ofZ^p so that I ∩H = {0}. Fixk ∈ Z^q so thatI (k) = I. Sinceϕ^∗()k = kfor all ∈ H and the Fourier series of any h∈C^∞(T^q)is absolutely convergent, see [14], the series

∈H

(h◦ϕ())(k)

is also absolutely convergent. Thus ρ_k^H(h)=

∈H

(h◦ϕ())(k), h∈C^∞(T^q) (6)

defines a distributionρ_k^H invariant by the restriction ofϕtoH i.e.

ρ_k^H(h◦ϕ())=ρ_k^H(h), for eachh∈C^∞(T^q)and for all∈H.

Moreover a routine computation shows that

ρ_k^H(h◦ϕ())=e^{2πi ϕ∗(−)k,α()}ρ_ϕ^H∗_(−)k(h), for all∈Z^p. (7) Thus if∈I,thenρ_k^H(h−h◦ϕ())=(1−e^{2πi k,α()})ρ_k^H(h).

We denote byρ_kthe distribution corresponding to the cyclic groupHgenerated by∈Z^p−I. This distribution is invariant byϕ().

Proposition 4.1. Letϕ be an affineZ^p-action onT^q withσ (ϕ∗) = {1}. If the first cohomology group ofϕis finite dimensional, thenϕsatisfies the irrationality condition.

Proof. Suppose thatϕdoes not satisfies the irrationality condition, then there existk∈Z^q− {0}so thatdimI (k)≥p−1 and

k, α() ∈Z, for all∈I (k). (8) We show that the first cohomology ofϕ is infinite dimensional. We first notice thatI (nk)=I (k)and nk, α() ∈Zfor alln∈Z, n=0.

We consider two cases.

Case 1. I (k)=Z^p.

We construct an infinite sequence of cocyclesfn, n∈Z− {0}giving linearly independent cohomology classes in the first cohomology group of ϕ. Since nk, ϕ()x − nk, xis integer for all ∈ Z^p and each ninZthenfn() = ¹cos 2π nk,·, = (¹, . . . , p) ∈ Z^p defines a cocycle for each n ∈ Z.

Moreover for any smooth functionh:T^q −→Rwe have that (h◦ϕ())(nk)=h(nk), for all∈Z^pand alln∈Z.

From this we see easily that the cocyclesfngive an infinite linearly independent sequence of cohomology classes.

Case 2. dimI (k)=p−1.

In this case there is, 1 ≤ i ≤ p so thatei ∈ I (k). Thus Z^p = I (k)⊕H whereH is the subgroup generated byei. Letρ_nk^ei as in(6)be the distributions invariant byϕ(ei). Now we choose functionshn∈C^∞(T^q),such thathn(k)=0 ifkdoes not belong to the orbitϕ^∗(j ei)nk, j ∈Zofnkfor eachn∈Z− {0}. We now define an infinite sequence of cocyclesfn ofϕ as followsfn(j ei) = hn+hn◦ϕ(ei)+ · · · +hn◦ϕ((j −1)ei), forj ≥ 0 andfn(j ei) = −(hn + hn◦ϕ(−ei)+ · · · +hn ◦ϕ(−(j −1)ei)), forj < 0 for all ∈ I (k). Thus fn(j ei)◦ϕ(k) = fn(j ei). We see from(8)that ϕ^∗()nk, α() =n k, α() is an integer. The mappingsfn :Z^p −→C^∞(T^q)given byfn() =fn(P ()) whereP is the projection ontoH,Z^p=I (k)⊕H are easily seen to be linearly independent cohomology classes in the first cohomology ofϕ.

Letϕ be an affineZ^p-action on the torusT^q. For each isotropy group ofϕ^∗ we consider the submoduleMI = {f ∈C^∞(T^q)|f (k) =0 for allk, I (k)=I}

ofC_ϕ^∞(T^q). For eachf ∈C^∞(T^q), fI denotes the projection off ontoMI. Lemma 4.2. Letϕ be an affineZ^p-action on the torusT^q andI = Z^p be an isotropy group of the actionϕ^∗. Suppose that for some∈I andf ∈C^∞(T^q) we have the equations

ρ_k(f )=0for allk∈Z^q, satisfyingI (k)=I. (9) Then there is a functionhI ∈MI so that

hI −hI ◦ϕ()=fI.

Moreover for eachr ≥1the Fourier coefficients ofhI satisfy the inequality

|hI(k)| ≤C(r)|k|^−r for allk, such thatI (k)=I where the constantC(r)dependent only onr, f and.

Proof. Consider the number given by(9)we have h(k)=

n≥0

(f◦ϕ(n))(k)= −

n≥1

(f ◦ϕ(−n))(k) (10)

for allk∈Z^q, I (k)=I.

Now we show that the Fourier series

k∈Z^q,I (k)=Ih(k)φk defines a function inMI i.e., for eachm >0 there existC(m) >0 such that|h(k)| · |k|^m≤C(m), for allk ∈ Z^q− {0}, I (k)= I. For eachk ∈ Z^q − {0}, I (k) =I there exist k0∈Z^qin the orbit ofkbyϕ^∗()such that

|ϕ^∗(n)k0| ≥ |k0|, for alln∈Z.

By Lemma A in the appendix there is a constantE >0 so that|ϕ^∗(n)k0| ≥E|n|

for alln ∈ Z. Considern⁰ ∈ Zsuch thatk =ϕ^∗(n⁰)k⁰. Sincef ∈ C^∞(T^q) then for eachs ≥ 0 there existC(s) > 0 so that|f (k)| · |k| ^s ≤ C(s). There two possibilities.

Ifn⁰≥0 we choose in(10)the equation h(k)= −

n≥1

(f ◦ϕ(−n))(k)=

n≥1

e^{2πi k,α(n)}f (ϕ ^∗(n)k).

Takings =m+r, wherer will be chosen later, we have that

|h(k)| · |k|^m≤C(m+r)

n≥1

|k|^m

|(ϕ^∗(n))k|^m+r =C(m+r)

n≥1

|ϕ^∗(n0)k0|^m

|ϕ^∗((n+n0))k0|^m+r.

Asσ (ϕ^∗())= {1}thenN()=ϕ^∗()−idis a nilpotent matrix, thusN()^d =0, for some 1≤d ≤q. Thus there isC⁰>0 so that for everyn⁰∈Z

|ϕ^∗(n0)| ≤C0max(1,|n0|^d−1).

Hence|ϕ^∗(n0)k0| ≤C0max(1,|n0|^d−1)|k0|and

|h(k)| · |k|^m ≤ C(m+r)C0^m

n≥1

max(1,|n0|^d−1)^m|k0|^m

|ϕ^∗((n+n0))k0|^m+r

≤ C(m+r)C0^mE⁻¹

n≥1

max(1,|n0|^d−1)^m (n+n0)^r .

Choosingr =m(d−1)+1+ε, for someε >0 we have that n^(d−0 ^1)m

(n+n0)^r ≤ 1 (n+n0)^1+ε.

Thus|h(k)| · |k|^m ≤C(m), where

C(m)=C(md+1+ε)C0^mE⁻¹

n≥1

1 n^1+ε depends onbut not onI.

The casen0<0 follows in a similar way considering that h(k)=

n≥0

f◦ϕ(n)(k)=

n≥0

e^{2πi k,α(−n)}fϕ^∗(−n)k.

From(10)we easily thathI −hI ◦ϕ()=fI.

5 Computing the first cohomology group

Lemma 5.1. Let ϕ : Z^p −→ Affine(T^q) be a minimal action and I be an isotropy group ofϕ^∗. Let∈I andh∈L¹(T^q,R). If∂h=h−h◦ϕ()=0 thenh(k)=0for allk∈Z^qsuch thatI (k)=I.

Proof. The equation∂h = 0 implies thath−h◦ϕ(n) = 0 for alln ∈ Z. Thus h(k)=e^{2πi ϕ∗(−n)k,α(n)}hϕ^∗(−n)k.

then since ∈I lim_n→∞|ϕ^∗(−n)k| = ∞and sinceh ∈ L¹(T^q,R)we con-

clude thath(k)=0.

Letf ∈Z¹(Z^p,kerm)be an 1-cocycle. We prove that for each isotropy group I ofϕ^∗the cocyclefI :Z^p −→MI, fI()=f ()I as in Lemma 4.2 is trivial, i.e., there ishI ∈MI so that

fI()=hI −hI ◦ϕ(), for all∈Z^p

wheref ()I denotes the projection off () ∈ kermonto MI. Moreover we prove that for allr ≥1 there isC(r) >0 independent ofI such that

|(hI)_k| ≤C(r)|k|^−r. Soh=

IhI defines a function inM0such that for all∈Z^p h−h◦ϕ()=f ().

This proves thatf ∈Z¹(Z^p,kerm)is a coboundary andH¹(Z^p,kerm)=0.

Proof of the Theorem 1. It remains to prove the implication 1⇒2. LetI be an isotropy group and let{e1, . . . , ep}be the canonical base ofZ^p. Letf be a 1-cocycle.

Case 1. I =Z^p. Fork∈existsuch thate^{2πi k,α()}=1. Define h(k)= f ()(k)

1−e^{2πi k,α()}. From the cocycle equation

f (+)=f ()+f ()◦ϕ()=f ()+f ()◦ϕ() we get

f ()−f ()◦ϕ()=f ()−f ()◦ϕ()for all∈Z^p thus f ()(k)(1−e^{2πi k,α()})=f ()(k)( 1−e^{2πi k,α()}) and we haveh(k)(1−e^{2πi k,α()})=f ()(k) .

By the Diophantine condition there existβ >0 andC >0 such that for each ofk∈,there isj,1≤j ≤pso that

k, α(ej) ≥C|k|^−β.

and for allr ≥1 there existC(r) >0 such that k, α(e_j) ≥ ^C(r)_|k|r >0.

Thush(k)= f (ej)(k)

1−e^{2πi k,α(ej)} and therefore

|h(k)| = |f (ej)(k)|

|1−e^{2πi k,α(ej)}| ≤ C(r)C⁻¹

|k|^r−β . Thus the functionhI =

h(k)φ(k) ∈ MI satisfieshI−hI ◦ϕ() =f ()I, for all∈I =Z^p, wheref ()I ∈MIis the projection off ()∈M0ontoMI. Case 2. dimI =p−1. Let 1≤ i≤ psuch thatei ∈I. By the, irrationality condition, for eachk ∈ Z^q − {0} such that I (k) = I there is ∈ I so that e^{2πi k,α()}=1. From the cocycle equation

f ()−f ()◦ϕ(ei)=f (ei)−f (ei)◦ϕ()

we getρ_k^ei(∂f (ei))=ρ_k(∂^eif ())=0. From(7)and the irrationality condi- tion we see thatρ_k^ei(∂f (ei)) =0. By Lemma 4.2 there ishI ∈ MI such that f (ei)I =hI −hI ◦ϕ(ei).

Projecting the cocycle equation ontoMI we get fI()−fI()◦ϕ(ei)=f (ei)I−f (ei)I◦ϕ()=

=hI −hI ◦ϕ()−(hI −hI ◦ϕ())◦ϕ(ei) Thus by Lemma 5.1 we see thatf ()I =hI −hI ◦ϕ()for all∈Z^p.

Note that Lemma 4.2 implies that for everyr ≥ 1 there is C(r) > 0, only depending inf (e1), . . . , f (ep)and independent ofI, such that

|hI(k)|< C(r)|k|^−r.

Case 3. dimI < p−1. There areei andej so thatH ∩I = {0},whereH is subgroup generated byei andej. Thus for eachk∈Z^q, I (k)=I we have that

ϕ^∗()k =kfor all∈H. (11)

From the cocycle equation

f (ej)−f (ej)◦ϕ(ei)=f (ei)−f (ei)◦ϕ(ej) a routine computation shows that

ρ_k^ei(f (ei))=ρ_k^ei(f (ei)◦ϕ(nej))for alln∈Z

from(7)we get

ρ_k^ei(f (ei))=e^{2πi ϕ∗(−nej)k,α(nej)}ρ_ϕ^ei∗(−nej)k(f (ei))for alln∈Z. (12) Since the series in (6) is absolutely convergent from (11) we see that

n→∞lim ρ_ϕ^ei∗(−nej)k(f (ei))=0.

Now from(12)we getρ_k^ei(f (ei))=0 and we proceed as in the Case 2 above.

In both case 2 and 3 Lemma 4.2 implies that for everyr ≥1 there isC(r) >0, only depending onf (e¹), . . . , f (ep)and independent ofI,such that

|hI(k)|< C(r)|k|^−r.

Appendix

Lemma A. LetA ∈ GL(q,Z)such thatσ (A) = {1}. Then there isC > 0 such that for alln ∈Zand allk ∈ Z^qsatisfyingAk= k, either|Aⁿk| ≥ C|n|

or|k| ≥C|n|.

Proof. By Jordan canonical form theorem there is aP ∈ M(q,Z) such that P AP⁻¹is in upper triangular Jordan form, which means there is a decomposition ofP AP⁻¹into a direct sums of Jordan blocks. It is enough to prove the lemma for each Jordan block.

IfJ is am×mJordan block then there is(m)so that

|1 j!(1− 1

n)· · ·(1−j −1

n )| ≥(m)for alln∈Z andj,1≤j ≤m−1 such thatn > j or equivalently

| n

j

| ≥(m)|n|^j for alln∈Zand 1≤j ≤m−1 such thatn > j. (13) TakeC = _m+1,suppose that|k| ≤C|n|for somek∈Z^q− {0}such thatJ k=k andn∈Z− {0},we show that|Jⁿk| ≥C|n|. In fact, the first coordinate ofJⁿk is equal to

k¹+nk²+ · · · + n

j

kj+ · · · + n

m−1

kmwherek=(k¹, . . . , km).

Letj, 0≤j ≤m−1 be the largest integer so that n

j

kj =0. SinceJ k =k thenj >0.

Note that,

|k1+ · · · + n

j−1

kj−1|< m|n|^j−1 ε

m+1|n| = m

m+1ε|n|^j. Now from the inequality(13)we get

|k1+k2+ · · · + n

j

kj| ≥ | n

j

kj| − m

m+1ε|n|^j ≥ ε m+1|n|.

Thus|Jⁿk| ≥ _m+^ε₁|n|and the lemma follows forJ withC= ^ε(m)_m+1.

Acknowledgment. I would like to thank J. Rivera-Letelier for valuable com- ments and remarks and to N.M. dos Santos for suggesting the original question and for suggestions and comments on the final version of this article. I’m also grateful to SUNY at Stony Brook and UFF at Niteroi, Brazil where part of this paper was written.

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Preprint 2001.

Richard Urzúa Luz

Universidad Católica del Norte Departamento de Matemáticas casilla 1280 Antofagasta CHILE

E-mail: [email protected]