CédricRousseau OnSobolevinfinitesimalrigidityoflinearhyperbolicactionsonthe2-torus*

Bull Braz Math Soc, New Series 36(3), 379-391

© 2005, Sociedade Brasileira de Matemática

On Sobolev infinitesimal rigidity of linear hyperbolic actions on the 2-torus*

Cédric Rousseau

Abstract. Let A be a symmetric hyperbolic matrix in SL(2,Z)and0the subgroup of SL(2,Z)generated by A. We aim to study the infinitesimal rigidity of the standard action of0on the torusT². More precisely, we will consider the Sobolev W^s–infinitesimal rigidity of this action (that is to determine if the cohomology space H¹(0,W^s(T M)) is trivial or not), and show that it is W^s–infinitesimally rigid only if 0 ≤ s < 1. A consequence will be that this action is not C^∞–infinitesimally rigid.

Keywords: infinitesimal rigidity, hyperbolic actions.

Mathematical subject classification: 22E40, 57S25, 58F15.

Introduction

Let G be a topological group and0 be a finitely-generated group. We denote by R(0,G) the set of all group homomorphisms of 0 into G endowed with the topology of pointwise convergence. If γ1, . . . , γk are fixed generators of 0, one may consider R(0,G) as a closed subset of G^k by means of the map ρ7→(ρ(γ1), . . . , ρ(γk)). Note that G acts naturally on R(0,G)by conjugation:

ifρ ∈ R(0,G) and g ∈ G, then for allγ ∈ 0, (g.ρ)(γ ) = gρ(γ )g⁻¹. A homomorphismρ0is said to be locally rigid if its orbit is open in R(0,G), or equivalently, if there exists a neighborhood U ofρ0in R(0,G)such that every ρ∈U is conjugated toρ0. In the case G is a Lie group, G acts differentiably on itself by conjugation (for any g0in G,8g₀ : g 7→g0gg0−1is an automorphism of G), and this action induces an action of G on its Lie Algebra g, which is isomorphic to the tangent space T1G, by means of the derivatives8⁰_g0(1). This is what is called the adjoint representation AdG of G in g. In this context,

Received 17 June 2004.

*I would like to thank A. El Kacimi for introducing me this problem about which we had many fruitful discussions.

Weil [W] proved that ifρ ∈ R(0,G)is such that H¹(0,AdG◦ρ)=0 thenρis locally rigid (see also Raghunathan [R], chapter VI).

Let now M be a compact C^∞manifold and Diff(M)be the group of C^∞diffeo- morphisms of M endowed with the usual C^∞topology. We denote by C^∞(T M) the space of C^∞ vector fields on M. Any representation ρ ∈ R(0,Diff(M)) induces a linear action of0on C^∞(T M), andρis said to be C^∞–infinitesimally rigid (or for short infinitesimally rigid) if H¹(0,C^∞(T M)) = 0. This termi- nology used by Zimmer [Z] suggests an analogy with Weil’s theorem. Indeed, C^∞(T M)is the Lie algebra of the infinite dimensional Lie group Diff(M), and the natural action of0 on C^∞(T M)is in fact the composition of the represen- tation of0 into Diff(M)with the adjoint representation of Diff(M) on its Lie algebra. Nevertheless, there is no established results connecting infinitesimal and local rigidity.

For results about local rigidity of the standard action of SL(n,Z)on the torus Tⁿ, the reader can refer to [H1], [H2], [KL], [KLZ]. With regard to infinitesimal rigidity of these actions on tori, many results have been established, especially when n≥3 and0is a subgroup of finite index in SL(n,Z). For instance, Pollicott [P] showed that the action of SL(3,Z)onT³is infinitesimally rigid, and Lewis [Le] proved that for n ≥ 7 and 0 a subgroup of finite index in SL(n,Z), the action of0onTⁿis also infinitesimally rigid. A more general result is given by Hurder [H3], stating that for n≥3 and0a subgroup of finite index in SL(n,Z), every affine action of0onTⁿassociated to the standard action is infinitesimally rigid. The reader interested by affine actions on tori can also refer to [Lu]. As a matter of fact the case n = 2 is raised in [H1], [H2], but these results only concern local rigidity.

The goal of this note is to study infinitesimal rigidity for the following example:

let A be a symmetric and hyperbolic matrix (that is to say that A has no eigenvalue of modulus 1, or equivalently in this case, that|tr A| > 2) in SL(2,Z)acting linearly on M = T². This matrix has two irrational eigenvalues, say λ and λ⁻¹ with|λ| >1. The infinite cyclic subgroup0 generated by A is of infinite index in SL(2,Z)and its action on M is Anosov. Let L²(T M)be the Hilbert space of square integrable vector fields on M and, for every real number s≥0, denote by W^s(T M) the space of vector fields of s–Sobolev class; naturally W⁰(T M)=L²(T M)and the intersection W^∞(T M)of all the W^s(T M)is equal to C^∞(T M). We will show that the action of0on M is W^s–infinitesimally rigid if and only if s < 1, and in particular, that the action of0 on M is not C^∞– infinitesimally rigid.

1 The space H⁰(0,L²(T M))of invariant L²vector fields

The problem of infinitesimal rigidity of the action of0on M, that is to determine H¹(0,C^∞(T M)), makes sense when considering real vector fields on M. How- ever, the special case of M=T²will enable us to use Fourier analysis and then to treat the more general problem of the cohomology of0 acting on complex vector fields on M.

The local coordinates of a particular point z ∈ M = T²will be denoted by (x,y). If X is a vector field on M, then on the covering spaceR², we have

X(z)= f(x,y) ∂

∂x +g(x,y) ∂

∂y ,

where f and g areZ²-periodic functions. The field X is said to be L² if the coefficients f and g are L² functions on M. In this case, f and g have the following Fourier expansions:

f(x,y)=X

m,n

fm,ne^2ıπ(mx+ny) , g(x,y)=X

m,n

gm,ne^2ıπ(mx+ny),

where(fm,n)and(gm,n)are elements of the Hilbert space`²(Z²,C)of complex square summable families indexed byZ². In the case X is a real vector field, coefficients fm,nand gm,nverify additional relations:

f₋m,−n= fm,n and g₋m,−n =gm,n.

From now on, we identify L²functions on M with elements of`²(Z²,C).

Since A= a b

b c

is symmetric, it admits a diagonalization in an orthogonal basis ofR², say

A= P

λ 0 0 λ⁻¹

P⁻¹ with P =

cosθ −sinθ sinθ cosθ

∈SO(2,R).

The action of0 =< A>on L²(T M)is given by A_∗X(Az)=(a f(z)+bg(z)) ∂

∂x +(b f(z)+cg(z)) ∂

∂y.

Linear eigenvector fields Xλ = cosθ_∂^∂_x +sinθ_∂y^∂ and X_λ−1 = −sinθ_∂^∂_x + cosθ_∂y^∂ respectively associated toλandλ⁻¹ (that is to say A_∗Xλ = λXλ and

A_∗X_λ−1 = λ⁻¹X_λ−1) form a basis of L²(T M)in which any vector field X is written X(z)=u(z)Xλ+v(z)X_λ−1, where u andvare L²functions on M. Then we have

A_∗X(Az)=λu(z)Xλ+λ⁻¹v(z)X_λ−1. (1) For convenience, we put for any(m,n)∈Z²:

m n

= A m

n

and m

n

= A⁻¹ m

n

.

If(m0,n0)is a reference pair inZ², we put for any k ∈Z: mk

nk

= A^k m0

n0

.

We can now show that

Theorem 1.1. The space H⁰(0,L²(T M)) of0–invariant L² vector fields is trivial.

Proof. An L² vector field X is invariant if and only if A_∗X = X ; this is equivalent to

∀z∈ M, u(Az)=λu(z) and v(Az)=λ⁻¹v(z).

Replacing for instance the function u by its Fourier expansion, we obtain:

X

p,q

up,qe^{2ıπ((ap+bq)x+(bp+cq)y)}=X

m,n

λum,ne^2ıπ(mx+ny)

i.e. X

p,q

up,qe^{2ıπ(px+q y)}=X

m,n

λum,ne^2ıπ(mx+ny).

By identifying coefficients, and doing it similarly forv, we obtain um,n=λum,nandvm,n=λ⁻¹vm,n.

Using these conditions, we can first deduce that u0,0=v0,0=0.

Suppose now that there exists a pair(m0,n0)6=0 such that um0,n0 6=0. Then for all k ∈ N, um_−k,n_−k = λ^kum₀,n₀; hence limk→−∞|um_k,n_k| = +∞. But this contradicts the fact that the orbit by A of every point ofZ²different from 0 is infinite (see lemma 1.3 below), and that the Fourier coefficient um,nof u tends to 0 when(m,n)tends to infinity. Thus we have um,n=0 (and in the same manner vm,n =0) for all(m,n)∈Z². So u=v=0 i.e. X =0.

We give from now on two simple lemmas concerning the orbits ofZ²for the action of A that will be useful especially in the last section of this paper. The first one is about the number of these orbits, and the second one specifies their asymptotic behaviour.

Lemma 1.2. There is a countable infinite number of orbits for the action of A onZ².

Proof. It suffices to remark that two pairs of integers that belong to the same orbit have the same G.C.D, so each point of the form(p,p)where p is a prime

number belongs to a unique orbit.

Lemma 1.3. For any fixed pair(m0,n0)6=0, there exists positive constants c₊ and c₋such that

m²_k+n²_k ∼

+∞c₊λ^2k and m²_k+n²_k ∼

−∞c₋λ^−2k

Proof. Since A=P

λ 0 0 λ⁻¹

P⁻¹with P=

cosθ −sinθ sinθ cosθ

, we have:

mk

nk

=P

λ^k 0 0 λ^−k

P⁻¹

m0

n0

=

λ^kcosθ (m0cosθ+n0sinθ )+λ^−ksinθ (−m0sinθ+n0cosθ )

−λ^ksinθ (m0cosθ+n0sinθ )+λ^−kcosθ (−m0sinθ+n0cosθ )

.

So

mk2+nk2=λ^2k(m0cosθ +n0sinθ )²+λ^−2k(−m0sinθ+n0cosθ )² thus

m²_k+n²_{k +∞}∼ c₊λ^2k, m²_k+n²_{k −∞}∼ c₋λ^−2k, where

c₊=(m0cosθ+n0sinθ )² and c₋=(−m0sinθ +n0cosθ )². Now(cosθ,sinθ )is an eigenvector of A and we can show then that cosθ and sinθ are rationally independent. So c₊>0 and c₋>0.

2 W^s-infinitesimal rigidity for 0≤s <1

For every real s≥0, we say that a vector field X is in W^s(T M), the s-Sobolev space of vector fields on M, if its coefficients are both in W^s(M), with

W^s(M)=n

f :Z²→C X

(m,n)6=0

(m²+n²)^s|fm,n|²<+∞o .

Then for 0≤s ≤s⁰, we have C^∞(M)=\

s≥0

W^s(M)⊂W^s0(M)⊂W^s(M)⊂W⁰(M)= L²(M).

For every s<∞, W^s(M)is a Hilbert space with the hermitian producth•,•i_s: hf,gi_s = f0,0g0,0 + X

(m,n)6=0

(m²+n²)^sfm,ngm,n,

and the associated norm :

||f||s =



|f0,0|²+ X

(m,n)6=0

(m²+n²)^s|fm,n|²





1 2

.

Let{δ^m,n}(m,n)∈Z² be the canonical Hilbert basis of`²(Z²,C), that is:

δ^m,n_p,q =

(1 if(p,q)=(m,n), 0 otherwise.

It is clear that{δ^m,n}is a Hilbert basis of W^s(M)for any s ≥0.

Let X ∈W^s(T M)with X(Z)=u(z)Xλ+v(z)Y_λ−1. According to expression (1), we have A_∗X(z)=U(z)Xλ+V(z)X_λ−1, with

U(z) = λu(A⁻¹z)=λX

m,n

um,ne^{2ıπ(m(cx−by)+n(−bx+ay))}

= λX

m,n

um,ne^{2ıπ(mx+n y)} =λX

m,n

um,ne^2ıπ(mx+ny); V(z) = λ⁻¹v(A⁻¹z)=λ⁻¹X

m,n

vm,ne^2ıπ(mx+ny).

So U =λTAu and V =λ⁻¹TAv, where TA is the linear operator consisting of the permutation of the indices of Fourier coefficients by A:

∀ f ∈ L²(M), (TAf)m,n= fm,n.

It is clear that TAis a bijective isometry of`²(Z²,C)and that TA−1=T_A−1. Let us be more precise with the the following proposition:

Proposition 2.1. For any s≥0, TAand T_A−1 are bijective continuous operators of W^s(M)and|||T_A−1|||_s = |||TA|||_s = |λ|^s.

Proof. Let f =X

m,n

fm,nδ^m,n ∈W^s(M); then TAf =X

m,n

fm,nδ^m,nand

||TAf||_s² = |f0,0|² + X

(m,n)6=0

(m²+n²)^s|fm,n|²

= |f0,0|² + X

(m,n)6=0

(m²+n²)^s|fm,n|².

Since A⁻¹is diagonalizable in an orthogonal basis, we can easily show that, as a linear operator of the usual euclidian spaceR²,|||A⁻¹||| = |λ|> 1. Conse- quently,

||TAf||_s²≤ |f0,0|² + X

(m,n)6=0

|||A⁻¹|||^2s(m²+n²)^s|fm,n|²

≤λ^2s||f||_s²,

and this for any f ∈W^s(M), hence|||TA|||_s ≤ |λ|^s.

Let > 0; the function x 7→ x^s is continuous in ]0,+∞[, therefore there existsη >0 such that

∀x >0, |x− |λ|| ≤η⇒ |x^s− |λ|^s| ≤. Then, for a small enoughη >0 :

|λ|^s−≤(|λ| −η)^s.

Now, as|||A⁻¹||| = |λ|, there exists a pair(p,q)∈Z×N^∗such that

|λ| −η≤ ||A⁻¹(p(p²+q²)⁻¹²,q(p²+q²)⁻¹²)|| ≤ |λ|, and so

(p²+q²)^s(|λ| −η)^2s ≤(p²+q²)^s ≤(p²+q²)^sλ^2s. Let g=(p²+q²)^−2sδ^p,q ∈W^s(M), so that||g||_s =1 and

(|λ| −η)^2s ≤ ||TAg||_s²=(p²+q²)^s(p²+q²)^−s ≤λ^2s,

and then

|λ|^s − ≤(|λ| −η)^s ≤ ||TAg||_s ≤ |λ|^s. Thus,

∀ >0, ∃g ∈W^s(M), ||g||_s =1 and |λ|^s−≤ ||TAg||_s ≤ |λ|^s, and we can conclude that |||TA|||_s = |λ|^s. Moreover, all this holds for A⁻¹ instead of A, so that we also have|||T_A−1|||_s = |λ|^s. We can then assert:

Corollary 2.2. For any s ≥0, W^s(T M)is a sub–0–module of L²(T M).

Since the group0 is isomorphic toZ, it is well known that

H¹(0,W^s(T M))=W^s(T M) A_∗X−X |X ∈W^s(T M) .

To prove that H¹(0,W^s(T M))= 0 is equivalent to show that, for each vector field Y in W^s(T M), there exists a vector field X in W^s(T M)such that A_∗X−X = Y . Let X(z)=u(z)Xλ+v(z)X_λ−1 and Y(z) =U(z)Xλ+V(z)X_λ−1; then the equation A_∗X−X =Y is equivalent to the system

(λTA−Id)u = U

(λ⁻¹TA−Id)v = V (2)

or equivalently:

(T_A−1 −λId)u = −T_A−1U

(TA−λId)v = λV (3)

Let us put SA =TA−λId and SA−1 =TA−1 −λId.

We can now easily prove that

Proposition 2.3. For 0≤s <1, H¹(0,W^s(T M))=0.

Proof. It is an established fact that, for a continuous operator of a Hilbert space (and more generally of a Banach space), its spectral radius is not greater than its norm. So, if 0 ≤ s < 1, then|λ|^s < |λ|and as a consequence, λis a regular value of both TAand T_A−1, that is to say SAand S_A−1 are invertible operators of W^s(M). Hence there exists a unique pair of functions u andvverifying system (3), and so the equation A_∗X−X =Y has a unique solution in W^s(M).

It is straightforward that TA and TA−1, and then SAand SA−1, sends real-valued functions to real-valued functions. It is not much more difficult to show it for their inverses S⁻_A¹and S⁻_A−¹1. So we can assert:

Theorem 2.4. For 0≤s <1, the action of0on M is W^s–infinitesimally rigid.

3 The space H¹(0,W^s(T M))for s ≥1

Now, let us show that, for s ≥ 1, the vector space H¹(0,W^s(T M)) is non trivial. In order to do that, we just have to exhibit a vector field Y in W^s(T M) such that the unique solution X in L²(T M)of the equation A_∗X−X =Y is not in W^s(T M). It is then useful to determine the preimages of the basis{δ^m,n}by S_A−1.

Proposition 3.1. For any fixed pair(m0,n0)∈ Z², we setη^m0,n0 =S⁻_A−¹₁δ^m0,n0. Then

η^0,0= 1

1−λδ^0,0 and η^m0,n0 = − X+∞

k=0

λ^−k−1δ^mk,nk if (m0,n0)6=0.

Proof. By using the relationη^m_m,n^0,n0 −λη^m_m,n^0,n0 =δ_m,n^m0,n0, we have:

• If(m0,n0)=0, then

(η^0,0_0,0−λη_0,0^0,0=1

η^0,0_m,n−λη^0,0_m,n =0 if(m,n)6=0;

hence 





η_0,0^0,0= 1 1−λ,

η_m,n^0,0 =λη_m,n^0,0 if(m,n)6=0.

If we refer to the proof of theorem 1.1, we know that for(m,n)6= 0, we necessarily haveη_m,n^0,0 =0, hence

η^0,0= 1 1−λδ^0,0.

• If(m0,n0)6=0, then

(η_m^m0₋^,n_1,n⁰₋₁ −λη^m_m⁰₀^,n_,n⁰₀ =1

η_m,n^m0,n0−λη_m,n^m0,n0 =0 if(m,n)6=(m0,n0).

Thus

η^m_m^0,n0

−1,n₋₁ = 1+λη_m^m0,n0

0,n₀ ,

η^m_m⁰₋^,n_2,n⁰₋₂ = λη^m_m⁰₋^,n_1,n⁰₋₁ =λ(1+λη^m_m⁰₀^,n_,n⁰₀) , ...

η^m_m⁰₋^,n_k,n⁰_−k = λ^k−1(1+λη_m^m₀^0,n_,n0⁰) for all k ≥1.

Similarly, in the opposite direction, we have also η^m_m⁰₁^,n_,n⁰₁ = λ⁻¹η^m_m0^0,n_,n⁰₀ ,

η^m_m⁰₂^,n_,n⁰₂ = λ⁻²η^m_m0^0,n_,n⁰₀ , ...

η^m_m⁰_k^,n_,n⁰_k = λ^−kη^m_m0^0,n_,n⁰₀ for all k ≥0, Once again because limk→+∞η_m^m0,n0

−k,n_−k = 0, we necessarily have 1 + λη^m_m⁰₀^,n_,n⁰₀ =0, and consequently

η^m_m⁰_k,n^,n_k⁰ =

(0 if k <0,

−λ^−k−1 if k ≥0.

When the point(m,n) does not belong to the orbit of(m0,n0), we use the same argument again to assert that η^m_m,n^0,n0 = 0. Finally, for all (m0,n0)6=0:

η^m0,n0 = − X+∞

k=0

λ^−k−1δ^mk,nk.

We verify by the way thatη^m0,n0 is actually in W⁰(M)=L²(M)and that

||η^m0,n0||₀ = |λ|

√1−λ⁻².

For every pair(m0,n0), it is clear thatδ^m0,n0is in W^s(M)for any real s; but what aboutη^m0,n0 (of course in the case(m0,n0) 6= 0, since it is obvious thatη^0,0 is in every W^s(M))?

Proposition 3.2. For any pair(m0,n0)6=0,η^m0,n0 is not in W^s(M)if s≥1.

Proof. We have to show that the seriesP

k≥0(m²_k +n²_k)^sλ^−2k−2is divergent.

Lemma 1.3 implies that(m²_k +n²_k)^sλ^−2k−2 _+∞∼ λ⁻²c₊^sλ^2(s−1)k, and the series P

k≥0(m²_k +n²_k)^sλ^−2k−2is convergent if and only if s <1.

Finally, we are in a position to give an example of a countable family{Yp}of real vector fields on M such that Yp ∈ W^s(T M)for any s ≥ 0 and for which the unique field Xpin L²(T M)such that A_∗Xp−Xp =Yp is not in W^s(T M) if s ≥1.

Proposition 3.3. Let_Pbe the set of prime numbers and put for every p∈P, Yp(z)=2 cos((a+b)px +(b+c)py)Xλ.

Then for any p∈P, the unique vector field Xp ∈L²(T M)such that A_∗Xp−Xp=Yp

is not in W^s(T M)for s≥1.

Proof. It is obvious that Yp =UpXλwith Up =δ^−p,−p+δ^p,pis in W^s(T M) for any s≥0. Now Xp=upXλwith up = −S⁻_A−¹1TA−1Up= −(η^−p,−p+η^p,p), and if we set(m0,n0)=(p, p), we have

up= X+∞

k=0

λ^−k−1δ^−mk,−nk+ X+∞

k=0

λ^−k−1δ^mk,nk,

hence

||up||_s²=2 X+∞

k=0

(m²_k+n²_k)^sλ^−2k−2=2||η^p,p||_s²= +∞.

So up 6∈W^s(M)and then Xp6∈W^s(T M).

Proposition 3.4. The family {[Yp]}^p∈P is linearly independent in H¹(0,W^s(T M))when s≥1.

Proof. Let J be a finite subset of_Pand(μp)p∈J ∈C^J such that X

p∈J

μp[Yp] =0 in W^s(T M), s ≥1.

Then, there exists X ∈W^s(T M), X =u Xλ, such thatP

p∈JμpYp= A_∗X−X with u=P

p∈Jμpup, and

||u||_s²=X

p∈J

|μp|²||up||_s²≥ |μq|²||uq||_s² for any q ∈ J.

So, since||u||_s²<+∞and||uq||_s²= +∞, we necessarily haveμq =0 for any

q ∈ J .

As a conclusion, we have:

Theorem 3.5. The action of0 on M = T²is W^s–infinitesimally rigid if and only if 0≤s <1. Moreover, in case 1≤s ≤ ∞, the space H¹(0,W^s(T M))is infinite-dimensionnal.

We could expect to obtain similar results for A∈SL(n,Z)acting onTⁿwith n > 2 using the same kind of method. However, in dimension 2, the fact that the two eigenvalues of A are the inverse one of the other appears to be essential.

So a generalization in higher dimension seems to be difficult, except maybe in the case n is even, because there exist then symmetric hyperbolic matrices A in SL(n,Z)such that A and A⁻¹have the same eigenvalues.

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Cédric Rousseau LAMATH

Université de Valenciennes et du Hainaut-Cambrésis 59313 Valenciennes CEDEX 9

FRANCE

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