LOCAL RIGIDITY OF PARTIALLY HYPERBOLIC ACTIONS. II. THE ... · II. THE GEOMETRIC METHOD AND...

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LOCAL RIGIDITY OF PARTIALLY HYPERBOLICACTIONS. II. THE GEOMETRIC METHOD AND

RESTRICTIONS OF WEYL CHAMBER FLOWS ONSL(n, R)/Γ

DANIJELA DAMJANOVIC AND ANATOLE KATOK 1

Abstract. We consider the restriction α0,G of the Weyl chamberflow on SL(n, R)/Γ (where Γ is a cocompact lattice) to a closedsubgroup G isomorphic to Zk × Rl, k + l ≥ 2 of the group D+

of positive diagonal matrices which contains a lattice in a two-dimensional plane in general position. We prove that any C2 smallsmooth perturbation of the action α0,G is differentiably conjugateto a standard perturbation which arises from a perturbation of theembedding Zk × Rl → D+.

We introduce a new method in rigidity of actions of higher rankabelian groups based on the study of combinatorial structure ofthe the web of Lyapunov (unipotent) foliations. Insights from theclassical algebraic K-theory play a crucial role in establishing sta-bility properties of that web. The method has applications to otherclasses of partially hyperbolic algebraic actions and is complemen-tary to the other new analytic method which we introduced in thefirst paper of this series.

1. Introduction

1.1. Actions of higher rank abelian groups and rigidity. In thispaper we make an essential step in realization of the program aimingat showing modified local differentiable rigidity for a broad class of al-gebraic (homogeneous and affine) partially hyperbolic actions of higherrank abelian groups. For definitions and general background on par-tially hyperbolic dynamical systems see [24]; all necessary backgroundon algebraic actions can be found in [15]. We will also strongly rely ondefinitions, constructions and results from our earlier paper [3].

Here we restrict ourselves to few brief comments on the rigidity prob-lem. For a more detailed discussion of the program see the introductionto [5]. In that paper a new method combining harmonic analysis witha KAM-type iteration scheme has been introduced which allowed for

1Based on research supported by NSF grants DMS 0071339 and DMS 05055391

2 DANIJELA DAMJANOVIC AND ANATOLE KATOK

the first time to prove rigidity for a class of abelian actions where indi-vidual elements do not possess structural stability. Results of [5] havebeen announced in [4].

In the present paper we introduce a totally different and comple-mentary method based on geometry and combinatorics of invariantfoliations and using insights from algebraic K-theory as an essentialtool.

We consider algebraic actions of Zk×Rl, k+l ≥ 2. The most generalcondition which leads to various rigidity phenomena (cocycle rigidity,local differentiable rigidity, measure rigidity, etc) is the following :

(R) The group Zk × Rl contains a subgroup L isomorphic to Z2 suchthat for the suspension of the restriction of the action to L every ele-ment other than identity acts ergodically with respect to the standardinvariant measure obtained from Haar measure.

A detailed discussion of this condition and its implications can befound in [15].

In the case of an action of Zk, (k ≥ 2) by automorphisms of the torusconsidered in [5] condition (R) is the same as existence of a subgroupisomorphic to Z2 which acts (with the exception of identity) by er-godic automorphisms, or, equivalently, absence of non-trivial rank onefactors; see [12], Section 2.2, conditions (R) and (R′) correspondingly.

In the present paper we treat another representative case of par-tially hyperbolic algebraic actions, satisfying condition (R), namelyrestrictions of the Weyl chamber flow on SL(n, R)/Γ. In section 1.5we discuss other applications and potential scope of our method.

1.2. Weyl chamber flow on SL(n, R)/Γ and generic restrictions.Let X := SL(n, R)/Γ with n ≥ 4 and Γ a cocompact lattice inSL(n, R). Let

D+ = exp D+ = {diag(exp t1, . . . , exp tn) : (t1, . . . , tn) ∈ Rn,

n∑k=1

tk = 0}

be the group of diagonal matrices in SL(n, R) with positive entries.The action by left translations of D+ on X is the Weyl chamber flow(WCF) and we denote this action by α0. For 0 ≤ i 6= j ≤ n thehyperplanes in D+ defined by Hij = {(t1, . . . , tn) ∈ D+ : ti = tj}are Lyapunov hyperplanes for the action α0, i.e. kernels of Lyapunovexponents of α0. Elements of D+ \

⋃i,j

Hij are regular elements of the

action. Connected components of the set of regular elements are Weylchambers.

RIGIDITY OF RESTRICTIONS OF WEYL CHAMBER FLOWS 3

The smallest non-trivial intersections of stable foliations of variouselements of the action α0 are Lyapunov foliations. Leaves of each Lya-punov foliation are one-dimensional. Each regular element either ex-ponentially expands or exponentially contracts each of those leaves.

A two-dimensional plane P ⊂ D+ is in general position if it intersectsany two distinct Lyapunov hyperplanes along distinct lines.

Let G ⊂ D+ be a closed subgroup which contains a lattice L in aplane in general position and let G = exp G.

One can naturally think of G as the image of an injective homomor-phism i0 : Zk × Rl → D+ (where k + l ≥ 2). Restriction of the WCFto a subgroup G will be denoted by α0,G. The action α0,G is given by

(1.1) α0,G(a, x) = i0(a) · xand will be referred to as a higher-rank generic restriction or justgeneric restriction of the WCF.

1.3. Cocycles and rigidity. Let α : A × M → M be an actionof a topological group A on a compact Riemannian manifold M bydiffeomorphisms. For a topological group Y a Y -valued cocycle (or anone-cocycle) over α is a continuous function β : A×M → Y satisfying:

(1.2) β(ab, x) = β(a, α(b, x))β(b, x)

for any a, b ∈ A. A cocycle is cohomologous to a constant cocycle(cocycle not depending on x) if there exists a homomorphism s : A →H and a continuous transfer map H : M → Y such that for all a ∈ A

(1.3) β(a, x) = H(α(a, x))s(a)H(x)−1.

In particular, a cocycle is a coboundary if it is cohomologous to thetrivial cocycle π(a) = idY , a ∈ A, i.e. if for all a ∈ A the followingequation holds:

(1.4) β(a, x) = H(α(a, x))H(x)−1

For more detailed information on cocycles adapted to the present set-ting see [3].

Let P ∈ D+ be a 2-dimensional plane in general position. We provedin [3] that every Holder cocycle with values in Rm over the action α0,G,where G is any subgroup of D+ which contains exp P, is cohomologousto a constant cocycle.

Unlike proofs of previous cocycle rigidity results for algebraic ac-tions of abelian groups, the proofs in [3] do not use harmonic analysis.Rather, we use the geometric structure of Lyapunov foliations of theaction. By applying the method of [13] we show in [3] that a cocy-cle over a partially hyperbolic action with locally transitive Lyapunov


foliations is cohomologous to a constant cocycle if and only if the peri-odic cycle functional (PCF) vanishes on all closed broken paths whosepieces lie in leaves of Lyapunov foliations of the action. Further, thepresentation of SL(n, R) [25] as well as the presentation of the kernelof its universal central extension [21], give explicit description of closedbroken paths along Lyapunov foliations which leads to vanishing of thePCF functional on all such paths and to cocycle rigidity. Smoothnessof the transfer map for smooth cocycles is a consequence of the factthat for a generic restriction of the WCF the Lyapunov distributionsalong with their Lie brackets generate the tangent space at every point.

1.4. Cocycle rigidity for perturbations and local rigidity; for-mulation of results. In this paper we notice first that the methodof [3] extends to actions α0,L where L = exp L and L is a lattice in aplane in general position (and of any subgroup of D+ which containssuch a lattice). Our main novelty is that the method extends also tosufficiently small C2 perturbations of a generic restriction α0,G. Thisallows us to obtain a homeomorphism conjugating a C2-small smoothperturbation of α0,G to α0,G. Smoothness of the conjugacy is then ob-tained by a general a priori regularity method as in [17] and [11]. Seealso [5, Section 2.1] for the description of that method in the context ofthe general rigidity program for actions of higher rank abelian groups.

Let α0,G be a higher rank generic restriction of WCF with the actinggroup Zk × Rl, k + l ≥ 2. Our main result is the following theorem.

Theorem 1. (Differentiable rigidity of generic restrictions) If αG isC∞ action of Zk × Rl sufficiently C2-close to α0,G, then there exists ahomomorphism i : Zk×Rl → D+ close to i0 and a C∞ diffeomorphismh : X → X such that α(a, h(x)) = h(i(a) · x) for all a ∈ Zk × Rl.

The principal ingredient in the proof of Theorem 1 is the next theo-rem which is the main technical result of the present paper. It gener-alizes the cocycle rigidity result from [3] to C2-small perturbations ofmore general generic restrictions.

Theorem 2. (Cocycle rigidity for perturbations) Let αG be a suffi-ciently C2-small C∞ perturbation of α0,G. If β is a Holder one-cocycleover αG with values in Rk, then β is cohomologous to a constant cocyclegiven by a homomorphism s : Zk ×Rl → D+ via a continuous transferfunction. Furthermore, if the cocycle β is sufficiently small in a Holdernorm the transfer map is C0 arbitrary small.

Other essential ingredients in the proof of Theorem 1 are Hirsch-Pugh-Shub stability theory [6], Theorem 3 from Section 6 below which


describes holonomy of the neutral foliation of the perturbed actionalong the Lyapunov foliations, and the old “a priori regularity” methodfor smoothness of the conjugacy. We prove Theorem 1 using theseingredients in the next section. The rest of the paper is dedicated tothe proof of Theorem 2 and Theorem 3.

1.5. Scope of the method. Our method is applicable to partiallyhyperbolic algebraic actions of higher rank abelian groups which havesufficiently rich non-commutative structure of the web of Lyapunov fo-liations. The key point (in addition to local transitivity of Lyapunovfoliations) is that every Lyapunov cycle can be approximated by acomposition of conjugates to stable cycles. Checking this conditionin a specific situation requires understanding of combinatorics of Lya-punov foliations and has to be done separately for various cases. In thispaper we consider a very important representative case where classicalresults from algebraic K-theory provide necessary algebraic informa-tion. Restrictions of Weyl chamber flows for other classical split simpleLie groups are treated in [2]. Extension to non-split groups requiresinformation about generators and relations not readily available fromthe literature. The work on this case is in progress.

Cocycle rigidity for α0,G also holds for “sufficiently small” cocycleswith values in Lie groups [3, 16] and diffeomorphism groups [16]. Theseresults with suitable assumptions on the range can be extended tosmall perturbations of those actions in the way very similar to theproof of Theorem 2. They in turn produce the basis of proving localdifferentiable rigidity for a certain class of compact group and isometricextensions of Weyl chamber flows and their restrictions. Those resultswill appear in a subsequent paper.

2. Cocycle rigidity for perturbations implies localrigidity

2.1. Preliminaries from partial hyperbolicity theory.

2.1.1. Stability of neutral foliation. The neutral foliation for a genericrestriction α0,G is the orbit foliation of the WCF. Since it is a smooth fo-liation, we may use the Hirsch-Pugh-Shub structural stability theorem[6, Chapter 6]. Namely if αG is a sufficiently C1-small perturbationof α0,G then for all elements a ∈ A which are regular for α0,G andsufficiently away from non-regular ones (denote this set by A), the dif-feomorphism α(a, ·) is also partially hyperbolic, i.e. uniformly normallyhyperbolic. The central distribution is the same for any a ∈ A and isuniquely integrable to an α(a, ·)-invariant foliation which we denote by


N . Moreover, there is a Holder homeomorphism h of X, C0 close tothe idX , which maps leaves of N0 to leaves of N : hN0 = N . Thishomeomorphism is uniquely defined in the transverse direction, i.e. upto a homeomorphism preserving N . Furthermore, h can be chosensmooth and C1 close to the identity along the leaves of N0 althoughwe will not use the latter fact.

2.1.2. Holder conjugacy to perturbations along the leaves of the neutralfoliation of a generic restriction. Let us define an action αG of G on Xas the conjugate of αG by the map h obtained from the Hirsch-Pugh-Shub stability theorem:

αG := h−1 ◦ αG ◦ h

Clearly the leaves of the foliation N0 are preserved by every a ∈ A.The action αG is Holder but it is smooth and C1-close to α0,G alongthe leaves of the neutral foliation N0.

2.2. Proof of Theorem 1. Let αG be a C∞ action of Zk×Rl close toα0,G in C2 topology and let αG be the conjugate of αG obtained via the

Hirsch-Pugh-Shub homeomorphism h as explained in Section 2.1.2.Since the action αG is a C0 small perturbation of α0,G along the leaves

of the neutral foliation of α0,G whose leaves are {D+ · x : x ∈ X}, wehave that αG is given by a map β : (Zk × Rl)×X → D+ by

αG(a, x) = β(a, x) · α0,G(a, x)

for a ∈ Zk × Rl and x ∈ X. We will use multiplicative notation forthe abelian group D+ although it is isomorphic to Rn−1 and we willapply Theorem 2 to cocycles with values in D+. Notice that since αG

is a small perturbation of the action by left translations α0,G, it can belifted to a G-action on SL(n, R) commuting with the right Γ action onSL(n, R), and β is a cocycle over αG (for more details see [20, example2.3]). In particular we have:

(2.1) β(ab, x) = β(a, αG(b, x))β(b, x).

From this and Section 2.1.1 it follows that since αG is Holder, β(a, x)is small Holder cocycle over the action αG, due to the smallness ofthe perturbation. Thus by Theorem 2, β is cohomologous to a smallconstant cocycle s : Zk × Rl → D+ via a continuous transfer mapH : X → D+ which can be chosen close to identity in C0 topology ifthe perturbation αG small in C2 topology.


Let us consider the map h′(x) := H−1(x) · x. We have from thecocycle equation (2.1) and the cohomology equation (1.3)

h′(αG(a, x)) = α0,G(a, h′(x))

where α0,G(a, x) := i(a) · x, where i(a) := s(a)i0(a), a ∈ A and i0 is

as in (1.1). Since the map h′ is C0 close to the identity it is surjectiveand thus the action αG is semi-conjugate to the standard perturbationα0,G of α0,G., i.e. α0,G is a factor of αG.

Proposition 2.1. The map h′ is a homeomorphism and hence providesa topological conjugacy between αG and α0,G.

Proof: It is enough to prove that h′ is injective. Since the map h′

preserves leaves of the foliation N0 pre-image of any point belongs toa single leaf of that foliation. Furthermore, since h′ is close to identitythe diameter of each such pre-image is small.

Now we pass to the (almost universal) cover SL(n.R). Since themap h′ is close to the identity on X it is uniquely lifted to a close tothe identity map on the cover for which we will use the same notationh′. Furthermore, if we show that the lifted map is injective on thecover it will follow that the original h′ : X → X is injective and hencea homeomorphism.

Next we show that if h′(x) = h′(y) and H is an F holonomy, i.e aproduct of holonomy maps between leaves ofN0 within leaves of variouscenter-Lyapunov foliations Wij then h′(H(x)) = h′(H(y)). For thedefinition and discussion of holonomies see Section 6. Obviously it issufficient to prove this for a holonomyH within a single leaf ofWij. Butthis follows immediately from the fact that the semi-conjugacy mapscontracting manifolds of elements of αG into contraction manifolds ofcorresponding elements of α0,G.

Since the F holonomy group acts transitively on the leaves of N0 ifh′(x) = h′(y) there is an F holonomy map F of the leaf N x

0 of N0 suchthat F (x) = y. Hence h′(F n(x)) = h′(x) for any integer n, i.e. h′ mapsthe whole F orbit of x to the same point. But by Corollary 6.1 suchorbits cannot have compact closure in the topology of the leaf. Thiscontradiction proves that h′ is injective. �

Now by letting h := h′h−1 we have:

h ◦ αG ◦ h−1 = α0,G

thus there is a topological conjugacy between αG and a standard pertur-bation of α0,G. In particular, the conjugacy takes Lyapunov foliationsof α0,G into those of αG


Proving further that the conjugacy is smooth along the leaves of Lya-punov foliations of α0,G follows by an application of the Katok-Spatziermethod of non-stationary normal forms (see [17, Corollary 10 and Sec-

tion 2.2.2] or [20, Lemma 5.12 and Section 5.3]). The smoothness of hthen follows from the fact that Lyapunov directions for α0,G i.e. unipo-tent directions of the WCF, with their Lie brackets span the tangentspace at every point (see [18, Theorem 2.1] or [20, Theorem 5.9]).

3. Lyapunov cycles and cocycle rigidity partiallyhyperbolic actions

With a view of future applications material in this section is pre-sented in greater generality than is necessary for our immediate pur-pose. In particular for generic restrictions of the Weyl chamber flow onSL(n, R)/Γ and their perturbations all Lyapunov exponents are sim-ple and there are no positively proportional Lyapunov exponents; hencecoarse Lyapunov foliations coincide with Lyapunov foliations and areone-dimensional.

3.1. Preliminaries. Let α : A → Diff(M) be an action of A := Zk×Rl

on a compact manifold M by diffeomorphisms of M preserving anergodic probability measure µ. Then there are finitely many linearfunctionals λ on A, called Lyapunov exponents, a set of full measure Λand a measurable splitting of the tangent bundle TΛM =

⊕λ

Eλ, such

that for v ∈ Eλ and a ∈ A the Lyapunov exponent of v with respectto α(a) is λ(a).

If χ is a non-zero Lyapunov exponent then we define its coarse Lya-punov subspace by

Eχ :=⊕

{λ=cχ:c>0}

Eλ.

For every a ∈ A one can define stable, unstable and neutral subspacesfor a by: Es

a =⊕

λ(a)<0

Eλ, Eua =

⊕λ(a)>0

Eλ and E0a =

⊕λ(a)=0

Eλ. In

particular, for any a ∈ A :=⋂

χ6=0

(Kerχ)c the subspace E0a is the same

and thus can be denoted simply by E0. Hence we have for any such a:

TM = Esa ⊕ E0 ⊕ Eu

a .

See [9, Section 5.2] for more details.If in addition E0 is a continuous distribution uniquely integrable to

a foliation N with smooth leaves, and if there exists a ∈ A such thatα(a) is uniformly normally hyperbolic with respect to N (in the senseof the Hirsch-Pugh-Shub [6]) then α is a partially hyperbolic action.


Elements in A which are uniformly normally hyperbolic with respectto N are called regular. Let A be the set of regular elements.

If the set A is dense in Rk, then for each non-zero Lyapunov exponentχ and every p ∈ M the coarse Lyapunov distribution is:

Eχ(p) =⋂

{a∈A:χ(a)<0}

Esa(p).

The right-hand side is Holder continuous and Eχ can be extended to aHolder distribution tangent to the foliation Tχ :=

⋂{a∈A:χ(a)<0}

Wsa with

C∞ leaves. This is the coarse Lyapunov foliation corresponding to χ(See [3, Section 2] and [10]).

We denote by χ1, .., χr a maximal collection of non-zero Lyapunov ex-ponents that are not positive multiples of one another and by T1, ..., Tr

the corresponding coarse Lyapunov foliations.Given a foliation Ti and x ∈ M we denote by Ti(x) the leaf of Ti

through x.

3.2. Paths and cycles for a collection of foliations. Let T1, . . . , Tr

be a collection of mutually transversal continuous foliations on M , withsmooth simply connected leaves.

Definition 1. For N ∈ N and jk ∈ {1, . . . , r}, k ∈ {1, . . . , N − 1}an ordered set of points p(j1, . . . , jN−1) : x1, . . . , xN ∈ M is called anT -path of length N if for every k ∈ {1, . . . , N − 1}, xi+1 ∈ Tjk

(xk).

Definition 2. For N ∈ N and jk ∈ {1, . . . , r}, k ∈ {1, . . . , N} anordered set of points c(j1, . . . , jN) : x1, . . . , xN , xN+1 = x1 ∈ M is calleda T -cycle of length N if for every k ∈ {1, . . . , N}, xk+1 ∈ Tjk

(xk). AT cycle which consists of a single point is a trivial T -cycle.

Remark. We will denote a T -cycle c(j1, . . . , jN) by c whenever theshort notation causes no confusion.

Let C(T ) denote the collection of T -cycles. For x ∈ X let Cx(T )the collection of T -cycles with an initial point x. In Cx(T ) cyclesc(j1, . . . , jN) : x1, . . . , xk, xk, . . . xN , xN+1 = x1 andc(j1, . . . , jN) : x1, . . . , xk, . . . xN , xN+1 = x1 are identified. In particulara cycle x, x, x, . . . , x is identified with the cycle o : x, which is called atrivial cycle.

Now we introduce natural operations in C(T ) modeled on the op-erations on loops which appear in the definition of the fundamentalgroup.


• For two cycles c(j1, . . . , jN) : x = x1, . . . xN , xN+1 = x andc′(j′1, . . . , j

′N) : x = x′1, . . . , x

′N , x′N+1 = x in Cx(T ) define their

composition or concatenation c ∗ c′ by

c ∗ c′(j1, . . . , jN , j′1, . . . , j′N) : x = x1, . . . xN , x′1, . . . , x

′N , x′N+1 = x

• The inverse of a cycle c := c(j1, . . . , jN) : x1, . . . , xN , xN+1 = x1

is the cycle

c−1 := c−1(jN , . . . , j1) : x1, xN , . . . , x2, x1

.

• Let c(j1, . . . , jN) : x1, . . . , xN , xN+1 = x1 ∈ M be a T -cycle andlet y1 ∈ Tj(x1) for some j ∈ {1, . . . , r}. Then we call the cycle

c′(j, j1, . . . , jN , j) : y1, x1, x2, . . . , xN , xN+1 = x1, y1

a conjugate (or a Tj-conjugate) of c(j1, . . . , jN).

• Let c(j1, . . . , jN) : x1, x2, . . . , xN , xN+1 = x1 ∈ C(T ), and cn(j1, . . . , jN) :

x(n)1 , x

(n)2 , . . . , x

(n)N , x

(n)N+1 = x

(n)1 . Then c = limn→∞ cn if for all

k ∈ {1, . . . , N},xk = lim

n→∞x

(n)k .

Definition 3. A T -cycle c(j1, . . . , jN) : x1, . . . , xN , xN+1 = x1 ∈ Mis contractible if there is a closed broken path c in M obtained byconnecting for each k = 1, . . . , N the points xk and xk+1 by a path onthe leaf of the foliation Tjk

, and c is contractible.

Notice that the class of contractible cycles is closed under operationsdescribed above: composition, taking inverse, conjugation and takinglimit.

3.3. Stable cycles, allowable substitutions and reducible cy-cles. Now let T1, . . . , Tr be coarse Lyapunov foliations with smoothleaves of a partially hyperbolic action α on M . Notice that under thisassumption leaves of Ti for each i are simply connected, because everyloop within a leaf is mapped by a diffeomorphism into a loop of anarbitrary small diameter, and hence contractible, inside another leaf.

Definition 4. A T -cycle c(j1, . . . , jN) : x1, . . . , xN , xN+1 = x1 ∈ M iscalled stable for the A action α if there exists a regular element a ∈ Rk

such that xk ∈ Wsa(x1) for all k ∈ {1, . . . , N} i.e. if

N⋂k=1

{a : χjk(a) < 0} 6= ∅.


Simplest examples of stable cycles are cycles of the type c(j, j) :x, y, x, i.e. cycles contained in a leaf of some Tj foliation.

Notice that every stable cycle is contractible since it is mapped by adiffeomorphism to a cycle of arbitrarily small diameter.

Let ASsT (α) denote the collection of stable T -cycles.

Definition 5. A path p : x1, x2, . . . , xN−1, xN ∈ M reduces to a pathp′ : x1, x

′2, . . . , x

′N−1, xN ∈ M via an α-allowable T -substitution of s-

type if the T -cycle

p ∗ p′ : x1, x2, . . . , xN−1, xN , x′N−1, . . . , x′2, x1

obtained by concatenation of p and p′ is in the collection ASsT (α).

We denote by ASrsT (α) the collection of all T -cycles which reduce

to a trivial cycle i.e. to a point, via finitely many α-allowable T -substitutions of s-type. In particular, ASrs

T (α) contains conjugates ofall cycles in ASs

T (α)LetAST (α) denote the collection of T -cycles which containsASrs

T (α)and is closed in C(T ) (under the limiting procedure, Section 3.2).

Definition 6. A path p : x1, x2, . . . , xN−1, xN ∈ M reduces to a pathp′ : x1, x

′2, . . . , x

′N−1, xN ∈ M via an α-allowable T -substitution if the

T -cycle p ∗ p′ : x1, x2, . . . , xN−1, xN , x′N−1, . . . , x′2, x1 obtained by con-

catenation of p and p′ is in the collection AST (α) . Accordingly, cyclesin AST (α) are called α-allowable T -substitutions.

Remarks.

(1) We will sometimes use notation AST (α)x (corr. ASsT (α)x,

ASrsT (α)x) for T cycles inAST (α) (corr. ASs

T (α),ASrsT (α))with

initial point x.(2) Since leaves of Ti for all i ∈ {1, . . . , r} are simply connected,

cycles in AST (α) are contractible.(3) If there are no homoclinic pairs of points for T1, . . . , Tr with

respect to the action α i.e. if for any Ti, Tj ∈ T such thatχi(a)χj(a) < 0 for all regular a, Ti(x) and Tj(x) intersect only atx for any x ∈ M , then paths p(j1, . . . , jN−1) : x1, x2, . . . , xN−1, xN

and p′(j′1, . . . , j′N−1) : x1, x2, . . . , xN−1, xN can be reduced one to

the other, and as we are interested in reducibility classes only,they may be identified.

Definition 7. Two T -cycles c1 and c2 are α-equivalent if c1 reducesto c2 via a finite sequence of α-allowable T -substitutions. A T -cyclewe call α-reducible if it is α-equivalent to a trivial T -cycle i.e. if it can


be reduced to a point via finitely many α-allowable T -substitutions.Clearly, all T -cycles in AST (α) are T -reducible.

The set of T -cycles with an initial point x ∈ X Cx(T ) factored by therelation of α-equivalence we denote by R(α, T )x. It clearly has a groupstructure under the operation induced by concatenation of T -cycles atx.

The following lemma immediately follows from the fact that everyT - cycle of the type c : x, y, x where y ∈ Tj(x) for some j ∈ {1, . . . , r},is a stable T -cycle and thus is α-reducible.

Lemma 3.1. For every i ∈ {1, . . . , r} and x ∈ M , y ∈ Ti(x), thegroups R(α, T )x and R(α, T )y are isomorphic.

3.4. Transitivity of foliations.

Definition 8. A collection T of foliations T = {T1, . . . , Tr} is calledtransitive if there exist N ∈ N and R > 0 such that any two pointsx, y ∈ X can be connected by a T -path

p(j1, . . . , jN−1) : x1 = x, x2 . . . , xN = y

such that

xk+1 ∈ Tjk(xk) and dTjk

(xk)(xk+1, xk) < R.

The following is an easy consequence of Lemma 3.1.

Lemma 3.2. If T is a transitive collection of coarse Lyapunov folia-tions of a partially hyperbolic action α then for x ∈ M , R(α, T )x areall isomorphic and hence can be denoted by R(α, T ).

Proof: For x and y in M due to transitivity of the collection Tof foliations T1, . . . , Tr, there exists a T -path p(j1, . . . , jN−1) : x =x1, x2, . . . , xN−1, xN = y. Now, because of the invariance in Lemma3.1, we have that R(α, T )x

1 = R(α, T )x2 = · · · = R(α, T )xN . ThusR(α, T )x = R(α, T )y. �

Remark. As in the case of the fundamental group, isomorphism be-tween R(α, T )x and R(α, T )y need not be canonical.

Definition 9. Foliations T1, . . . , Tr are locally transitive if there existsN ∈ N such that for any ε > 0 there exists δ > 0 such that for everyx ∈ M and for every y ∈ BX(x, δ) (where BM(x, δ) is δ ball in M)there is a T -path p(j1, . . . , jN−1) : x = x1, x2, .., xN−1, xN = y in theball BM(x, ε) such that xk+1 ∈ Tjk

(xk) and dTjk(xk)(xk+1, xk) < 2ε.


In other words, any two sufficiently close points can be connected bya T -path of not more than N pieces of a given bounded length. Here,for a submanifold Y in M , dY (x, y) denotes the infimum of lengths ofsmooth curves in Y connecting x and y.

Notice that local transitivity implies transitivity, see [1].

Definition 10. Let 0 < p < 1. Foliations T1, . . . , Tr are locally p-Holder transitive if there exists N ∈ N, δ > 0 and a constant L > 0such that for every x ∈ M and every y ∈ BM(x, δ) there is an T -path p :(j1, . . . , jN−1)x = x1, x2, .., xN = y such that for all k ∈ {1, . . . , N − 1}we have:

dTjk(xk, xk+1) < LdM(x, y)p.

Remark. All three transitivity properties: transitivity, local transi-tivity, p-Holder local transitivity, defined in the above definitions, arepreserved under a Holder conjugacy. The same holds true for the groupR(α, T ) when it is well defined, i.e. when T is a transitive collectionof Lyapunov foliations for an action α.

3.5. Reducibility and cocycle rigidity. Two propositions in thissection are easy consequences of [3, Proposition 4] and our definitionof α-reducible cycles.

First, for the convenience of the reader we recall the definition of theperiodic cycle functionals (PCF) (for more details [13] or [3]). We givethe definition in the multiplicative form which holds for cocycles withnon-abelian range.

Definition 11. For a partially hyperbolic A-action α on a compactmanifold M with coarse Lyapunov foliations T1, .., Tr and for a cocycleβ : A × M → Y over α, where Y is a Lie group, we define H-valuedpotential of β as P j

a (y, x) = limn→+∞

β(na, y)−1β(na, x), χj(a) < 0

P ja (y, x) = lim

n→−∞β(na, y)−1β(na, x), , χj(a) > 0

where a is a regular element in A, j ∈ {1, . . . , r}, x ∈ M and y ∈ Tj(x).Now for any T -cycle c : x1, .., xN+1 = x1 on M , we can define the

corresponding periodic cycle functional:

(3.1) PCF (c)(β) =N∏

i=1

P j(i)a (xi, xi+1)(β).

It is proved in [3] that the expression for (PCF) does not depend onthe choice of a.


Two essential properties of the PCF which are crucial for our pur-pose are that PCF is continuous and that it is invariant under theoperation of moving cycles around by elements of the action α. Thelatter property has an immediate consequence that PCF vanishes on allstable cycles and all cycles which can be reduced to a trivial cycle viaα-allowable T -substitutions of s-type or rs-type. The former allows usto consider limits of cycles and implies that PCF vanishes on any cyclewhich can be reduced to a trivial one via α-allowable substitutions, i.e.on any α-reducible cycle.

Proposition 3.1. If T : T1, . . . , Tr is a locally transitive collection ofcoarse Lyapunov foliations for a partially hyperbolic A-action α andif the group R(α, T ) has no non-trivial homomorphisms into Rl thenevery Holder Rl valued cocycle over α is cohomologous to a constantcocycle via a continuous transfer function.

If the cocycle is smooth then the transfer map is smooth along theleaves of the coarse Lyapunov foliations in T .

Proof: Let Cx(T ) be the collection of T -cycles with initial pointx ∈ M . If f : Cx(T ) → Rl is a map such that:

(1) f(c ∗ c1) = f(c) + f(c1) for c, c1 ∈ Cx(T ), where ∗ denotesconcatenation of cycles at point x,

(2) f vanishes on all stable T -cycles,(3) f is continuous,

then, by definition of α-reducible T -cycles we have that f vanishes onall α-reducible T -cycles in Cx(T ).

Thus f defines a homomorphism from R(α, T ) into Rl which byassumption has to be trivial. Therefore f vanishes on all T -cycles inCx(T ) and since x is arbitrary, on all T -cycles.

Now we use [3, Proposition 4]. The periodic cycle functional (PCF)which appears in [3, Proposition 4] satisfies conditions (1), (2), and(3) (see [3, Section 3]), thus by the above analysis the periodic cyclefunctional vanishes on all T -cycles. This implies trivialization of Rl

valued cocycles over α by [3, Proposition 4]. �

The following Proposition treats a case which occurs in the proofof Theorem 2 when one can study R(α, T ) by studying α-reducibilityclasses within homotopy classes of T -cycles. If every class has finiteorder, then reducibility of cocycles follows.

Proposition 3.2. If T : T1, . . . , Tr is a locally transitive collection ofcoarse Lyapunov foliations of a partially hyperbolic A-action α. Sup-pose that


(1) If T -cycle c is contractible then there exists n ∈ N such that cn

is α-reducible.(2) π1(M) has no nontrivial homomorphism into Rl

then every Rl-valued Holder cocycle over α is cohomologous to a con-stant cocycle.

If the cocycle is smooth then the transfer map is smooth along theleaves of the foliations in T . If the cocycle is small on a compact setof generators of A, uniformly on M in some Holder norm, then thetransfer map is C0 close to the trivial one.

Proof: Since by assumption a power of any contractible T -cycle isα-reducible, and the (PCF) vanishes on all α-reducible cycles it followsthat the (PCF) vanishes on a power of any contractible T -cycle. Sinceby definition of (PCF) it takes concatenation of two cycles to the sumof their corresponding (PCF)s, it follows that (PCF) vanishes on ev-ery contractible T -cycle. Hence (PCF) induces a homomorphism fromπ1(X) into Rl. Since it has to be trivial by assumption, (PCF) is trivialon every T -cycle. Thus the claim follows from [3, Proposition 4].

In particular, if the cocycle is small on generators in some Holdernorm then due to local transitivity of T -foliations and from the com-putation in [3, Proposition 2] it is easy to see that the transfer mapobtained by this construction is C0 close to the trivial one. �

4. Generic restrictions of WCF as partially hyperbolicactions: Lyapunov cycles and allowable substitutions

4.1. Lyapunov foliations for α0 and α0,G. Let eij(t), 1 ≤ i, j ≤n, i 6= j be the standard unipotent one-parameter subgroups in SL(n, R)and let Uij be the corresponding homogeneous unipotent foliations ofX. For x ∈ X = SL(n, R)/Γ, the leaf of Uij through x is Ux

ij ={eij(t)x|t ∈ R}. These foliations are invariant under the action by lefttranslations of any element in D+ and are the coarse Lyapunov folia-tions for the action α0 i.e. for the full WCF. If P is a 2-plane in generalposition then the foliations Uij, 1 ≤ i, j ≤ n, i 6= j are also Lyapunovfoliations for α0,P . The leaves of Uij are intersections of the leaves ofstable manifolds of the action by different elements of P. See [3, Section5.2] for details. The same holds for the action by any regular latticein P and thus for any generic restriction α0,G. We denote by U thecollection Uij, 1 ≤ i, j ≤ n, i 6= j. The neutral foliation for a genericrestriction α0,G will be denoted by N0. It is the orbit foliation of thefull WCF.


Remark. In case of actions of maximal split Cartan on some otherhomogeneous spaces, for example SL(n, C)/Γ, neutral foliation of theaction as well as any generic restriction contains the orbit foliation ofthe Cartan action, but its leaves are larger and contain also the compactpart of the centralizer of the maximal split Cartan subgroup.

4.2. Local transitivity of Uij foliations. The following is a simpleconsequence of the fact that the foliations Uij are totally non-integrableof index 2 (i.e. the vector fields tangent to Uij, 1 ≤ i 6= j ≤ n andtheir Lie brackets already span the tangent space at any x ∈ X), see[1, Theorem 4.2] or [13, Proposition 1].

Proposition 4.1. Foliations Uij, 1 ≤ i 6= j ≤ n are locally 12- Holder

transitive.

4.3. Elementary Lyapunov cycles for α0,G. One consequence ofthe Steinberg description of central extensions [25] is the following pre-sentation result for SL(n, R):

Steinberg Theorem. The group SL(n, R) is generated by unipotentelements eij(t), t ∈ R, 1 ≤ i 6= j ≤ n subject to standard commutatorrelations:

(4.1) eij(t)eij(s) = eij(t + s)

(4.2) [eij(t), ekl(s)] =

1, j 6= k, i 6= l

eil(st), j = k, i 6= l

ekj(−st), k 6= j, i = l

where [·, ·] denotes the commutator; and the following relations:

(4.3) h12(t)h12(s)h12(ts)−1 = 1

where h12(t) := e12(t)e21(−t−1)e12(t)e12(−1)e21(1)e12(−1) for each t ∈R∗.

Each one of the relations in (4.1), (4.2) and (4.3) induces a U -cycleat every point x ∈ X. For example, the relation [e12(t), e34(s)] = 1induces the cycle c : x1, x2, x3, x4, x1 where the points of the cycle are:

x1 = x, x2 = e34(−s)x, x3 = e12(−t)e34(−s)x, x4 = e34(s)e12(−t)e34(−s)x

with the initial point x ∈ X.We call U cycles obtained from the presentation relations (4.1), (4.2)

and (4.3) elementary U-cycles.Elementary cycles obtained from relations (4.1), (4.2) we will also

call commutator cycles. Notice that every commutator cycle lies inside


a single stable leaf of a certain element of the WCF. For this reason,such cycles are stable cycles for the WCF (Section 3.3). Moreover,each commutator cycle is also stable for any generic restriction of theWCF. Namely, from [3, Proposition 7] every commutator cycle is stablefor a restriction of the WCF to a plane P in general position. Since alattice in P intersects all the Weyl chambers in P, and by assumption Gcontains such a lattice, it follows that every commutator path is stablefor a generic restriction α0,G.

Relations (4.3) induce elementary U -cycles starting at every x ∈ X,each consisting of 18 points. However, these U -cycles are not in thesame stable leaf for any element of the WCF. In particular, they arenot stable for any α0,G.

We call the U - cycles induced by relations (4.3) diagonal cycles.Even though diagonal cycles are not stable, their structure is well

understood. Namely, the collection of all diagonal cycles at a point xwith the operation of concatenation of cycles at a point x is related tothe group K2R. This group K2R has the following presentation (see forexample [3, Section 5.3] for formulation, [22, Chapter 11] for detailedexposition, or [21] for the original proof):

Matsumoto Theorem. The group K2R is generated by symbols{s, t}, s, t ∈ R∗ subject to relations:1. {s′, t}{s, t} = {s′s, t}, {s, t}{s, t′} = {s, tt′}2. {s, 1− s} = 1, s 6= 13. {s,−s} = 1

In general, any mapping from R∗×R∗ into an abelian group A takinga pair (s, t) ∈ R∗ × R∗ to an element {s, t} in A, so that the images{s, t} satisfy the assertions 1. and 2. of the Matsumoto Theorem aboveis called a Steinberg symbol. It is proved in [22, Theorem A.1.] thatevery Steinberg symbol which is continuous in s and t has only elementsof order 2 in the image. In particular, if A is isomorphic to Rl thenevery Steinberg symbol into A is trivial.

The following Lemma is crucial for the method developed in thispaper. Namely, it demonstrates that the diagonal cycles, i.e. thoseU -cycles induced by relations (4.3), even those of them which are nei-ther stable nor reducible to stable, are in fact limits of U -cycles whichare reducible to stable ones, at least those diagonal cycles which arecontractible.

Lemma 4.1. Each elementary U-cycle, if contractible, induces an α0,G-allowable U-substitution:

(1) Commutator U-cycles induced by relations (4.1) and (4.2) arein ASs

U(α0,G).


(2) Diagonal U-cycles induced by relations (4.3) with s, t not bothnegative, are in ASU(α0,G).

(3) Diagonal U-cycles induced by relations (4.3) with s, t both nega-tive, can be reduced via α0,G-allowable T -substitutions of rs-typeto a U-cycle induced by the relation {−1,−1}. If doubled, thosecycles are also in ASU(α0,G).

Proof: (1) From the assumption that α0,G is a generic restriction ofthe WCF (see [3, Proposition 7]) we have that ASs

U(α0,G) = ASsU(α0).

In particular, all commutator U -cycles induced by relations (4.1) and(4.2) are stable for α0,G and thus are contained in ASs

U(α0,G).(2) Each generator {s, t} of the group K2R induces at any given point

x ∈ X a U -cycle. Call it cx{s,t}. It is a U cycle which consists of 18 points

and is induced by the corresponding relation h12(s)h12(t)h12(st)−1 = 1

in (4.3) with the initial point x, where

h12(t) := e12(t)e21(−t−1)e12(t)e12(−1)e21(1)e12(−1)

for every t ∈ R∗.Relations 1., 2. and 3. in Matsumoto Theorem are proved by

using only relations (4.1), (4.2) and/or their conjugates, see for ex-ample [22, Chapter 11]. This implies that U -cycles cx

{s,t}{s,t′}{s,tt′}−1 ,

cx{s,t}{s,t′}{s,tt′}−1 , cx

{s,1−s} and cx{s,−s} are in ASrs

U (α0,G).

In particular, since cx{s,t}{s,t′}{s,tt′}−1 = cx

{s,t} ∗ cx{s,t′} ∗ cx

{s,tt′}−1 , this

implies that if cx{s,t} and cx

{s,t′} are in ASrsU (α0,G) then so is cx

{s,tt′}.

Similarly, if cx{s,t} and cx

{s′,t} are in ASrsU (α0,G) then so is cx

{ss′,t}.This combined with the fact that cycles cx

{3,−3} and cx{3,−2} are in

ASrsU (α0,G) (due to 2. and 3. of Matsumoto Theorem) implies that all

cx{3,(−3)i(−2)j} are in ASrs

U (α0,G). Since the sequence {(−3)i(−2)j}i,j is

dense, we have that c{3,t} for every t ∈ R∗ is a U -cycle which is a limitof elements in ASrs

U (α0,G), so it is in ASU(α0,G).Similarly, starting with cx

{4,−3} ∈ ASrsU (α0,G) and cx

{4,−4} ∈ ASrsU (α0,G)

(again due to 2. and 3. of Matsumoto Theorem) we get cx{4,t} is in

ASU(α0,G) for every t ∈ R∗. Thus every cx{3i4j ,t} is in ASU(α0,G) and

so is any limit, i.e. any cx{s,t}, for s > 0.

In a similar way one obtains cx{s,t} ∈ ASU(α0,G), t > 0. This proves

part (2).This proof is an analogue of Milnor’s proof in [22, Theorem A1] that

continuous Steinberg symbols over R∗ have order 2 . Proof of part (3)is very similar to the part (2) and we omit it. �


Remark. The Lemma above holds on the covering SL(n, R) and willbe later used in that context as well.

4.4. Lyapunov cycles for α0,G. Any U -path which consists of Npoints is determined by its initial point x and an ordered sequenceof unipotent elements eikjk

(tk), k = 1, . . . , N .Every contractible U -cycle is represented by a relation

(4.4) ei1j1(t1) . . . eiN jN(tN) = 1

in the group SL(n, R). Call this U -cycle u. Using the presentationof SL(n, R) from the Steinberg Theorem, every relation of the type(4.4) above, is a product of relations (4.1),(4.2), (4.3) and/or theirconjugates. If contractible, then the product involves elements withs and t not both negative from (4.3). Therefore, via elementary U -cycles u can be reduced to the trivial cycle i.e. to a point. FromLemma 4.1 all elementary U -cycles of this type and their conjugatesare in ASU(α0,G). Thus due to the fact that leaves of Uij-foliations aresimply connected (which is a consequence of the assumption that Γ isa cocompact lattice) we have the following:

Proposition 4.2. For a generic restriction α0,G all contractible U-cycles are α0,G-reducible and vice versa, hence R(α0,G,U) is isomorphicto π1(X).

Remark. By Proposition 3.2 and using Margulis Normal SubgroupTheorem [19], the above Proposition implies cocycle rigidity for genericrestrictions of the WCF. This remark is merely a reformulation of themain result in [3].

Now let αG be a C2-small smooth perturbation of a generic restric-tion α0,G. The goal of the next two sections is to establish that thestructure of Lyapunov cycles for αG is not very different from that ofα0,G.

5. Preliminaries on perturbations of genericrestrictions of WCF

5.1. Lyapunov foliations for perturbations of generic restric-tions. From this point on we will use the stability result of Hirsch-Pugh-Shub to conjugate the action αG to an action αG preserving theneutral foliation N0 of α0 via a Holder conjugacy. This has been ex-plained already in Sections 2.1.1 and 2.1.2. Thus αG is a Holder per-turbation of α0,G along the leaves of the neutral foliation N0 althoughit is smooth along the leaves of that foliation.


We denote the collection of Lyapunov foliations for αG by F . SinceαG and αG are conjugate via a Holder map, the stable and unstable, andthus the corresponding Lyapunov foliations, are dynamically defined forthe action αG. Denote the collection of Lyapunov foliations for αG byF . Notice that we do not know whether the leaves of those foliationsare smooth manifolds. However, every foliation in F is integrable withN0 and the resulting center-Lyapunov foliation does have smooth leavessince it coincides with the center-Lyapunov foliation for α0,G.

Now we consider Lyapunov foliations for perturbations which corre-spond to the unipotent foliations Uij for α0,G. Let 1 ≤ i 6= j ≤ n. Leta1, . . . , am be different elements within the Weyl chambers in G and

away from the Weyl chamber walls (i.e. in A) such that Uij =m⋂

k=1

Wsak

(i.e. such that χij(ak) < 0, where Wsak

is the stable foliation forα0(ak, ·) and χij Lyapunov exponents for the action α0,G). We then

denote the corresponding intersections of stable foliation for α by Fij

i.e. Fij =k=m⋂k=1

Wsak

are the corresponding Lyapunov foliations for the

action α. We also denote Lyapunov foliations h−1Fij for αG by Fij.Although as was already noticed we do not know whether foliations

Fij have smooth leaves they are simultaneously homeomorphic to folia-

tions Fij with smooth leaves. Thus we can apply all notions developedin Section 3 to the collection Fij. We will do that from now on.

5.2. Stability of local transitivity. In [1] Brin and Pesin show thatproperty of local transitivity of stable and unstable foliations of a par-tially hyperbolic diffeomorphism persists under C2-small perturbations.This implies:

Proposition 5.1.

(1) If αG is sufficiently C2-small perturbation of α0,G then Lya-punov foliations for αG are locally transitive.

(2) Lyapunov foliations of αG are locally transitive.

Proof: (1) essentially follows from Theorem 4.2 in [1] on stability oflocal transitivity in case of a single partially hyperbolic diffeomorphismand two foliations of high regularity (stable and unstable); the sameargument as in [1] implies stable local transitivity in case of severalsmooth foliations, which is the situation here (the unipotent foliationsof the action α0,G are smooth and locally transitive by Proposition 4.1).

(2) is an immediate corollary of the fact that αG and αG are conju-gate via a Holder homeomorphism (Remark after Definition 10). �


6. Holonomy and canonical projections

In this section we will move back and forth between the phase spaceX = SL(n, R)/Γ and its covering SL(n, R) and use the same notationin both cases for all the invariant foliations. Since α0,G and αG both liftto the covering, the notions of reducible cycles and of all the collectionsdefined in Section 3.3 such as ASU(α0,G) and ASF(αG) make sense andwe keep here the same notation.

Remark. Notice that SL(n, R) is not the universal cover for X but forn ≥ 3 it is almost so, namely, the universal cover is a double cover ofSL(n, R). We could use the universal cover in the subsequent discussionbut it would make essentially no difference so we stick to the morefamiliar SL(n, R).

6.1. U- and F-holonomies. The foliationsN0 and Uij on the coveringspace SL(n, R) integrate to an invariant foliation Wij with the productstructure. This foliation is also invariant for αG. Moreover, everyleaf of Fij inside this foliation intersects every leaf of N0 at a uniquepoint. Thus we can define U -holonomy (along the leaves of Uij) andF -holonomy (along the leaves of Fij) between different leaves of N0

within a leaf of Wij.Now we can consider products of holonomies for different i, j pairs.

In particular, for every U -path and every F -path with end-points on thesame leaf of N0, the product of those holonomies is a map of the leaf. Itfollows immediately from commutation relations in SL(n, R) that anyU -holonomy is a translation and hence depends only on the end-pointsin a leaf of N0. Since the collection of Lyapunov foliations Uij is locallytransitive, every translation appears as a holonomy of a leaf N0. Hencethe group of U -holonomies is an abelian group isomorphic to D+ andacting simply transitively on each leaf of N0.

Proposition 5.1 shows that F -foliations on X are locally transitive.Since we consider F -holonomies on the cover, we make use of the fol-lowing fact:

Lemma 6.1. Foliations Fij, 1 ≤ i 6= j ≤ n on SL(n, R) are transitive.

Proof: We need to show that any two points x and y in SL(n, R) canbe connected by an F -path. Since the foliations Fij are transitive onX, it suffices to show that for a set of generators of Γ there are closed F -paths on the compact manifold which represent those generators of thefundamental group. This is true for U -paths. Take the correspondingF -paths (canonical projections of the U -paths); they may not be closedbut are nearly so because the paths are of uniformly bounded length


and the perturbation is small. Then by local transitivity (Proposition5.1) this small gap can be filled by a small F -path. Concatenation ofthat path and the F -path obtained as a projection of a U path rep-resenting some generator γ ∈ Γ gives an F -path in SL(n, R) whichrepresents γ. �

As a result of our study of F -cycles (Section 6.4) we will prove thefollowing Theorem:

Theorem 3. The group of F-holonomies acts simply transitively oneach leaf of N0.

We postpone the proof of Theorem 3 to Section 7.1. Here we proveits Corollary:

Corollary 6.1. The group of F-holonomies does not have any compactsubgroups.

Proof: Since the group of F -holonomies is homeomorphic to Rn, itis isomorphic to a Lie group (by Montgomery-Zippin [23]). If KAN isthe Iwasawa decomposition of the Lie group of F -holonomies, then thegroup is homotopic to compact K and since the group of F -holonomiesis contractible, so is K. But any compact contractible topological groupis trivial (see e.g. [8]), thus K is trivial and the group of F -holonomiesis solvable. Since a solvable simply connected Lie group cannot havecompact subgroups [7, Theorem 2.3], the claim follows. �

6.2. Correspondence between Lyapunov paths for α0,G and Lya-punov paths for αG. Now we define continuous maps (depending ona given foliation and a leaf) which take leaves of Fij to leaves of Uij andvice versa within the same center-Lyapunov leaf of Wij on the coveringspace, as projections along the leaves of the neutral foliation N0. Thesemaps which we call canonical projections, will allow us further to mapF -cycles to (possibly open) U -paths and U -cycles to F -paths.

In particular, since the canonical projections take pieces of the stablefoliation for a regular element of one action to pieces of the stablefoliation for the same element of the other action, they take stable F-cycles to stable U -cycles and vice versa. This is the key observation forour study of correspondences between F -cycles and U -cycles. However,not all U -cycles, even contractible ones, are stable, see (4.3) in Section4 and remarks thereof.

In this section, using canonical projections and:1. algebraic structure of elementary U -cycles,


2. homogeneous structure of the neutral foliationwe relate the reducibility classes of U -cycles and the reducibility classesof F -cycles. Since by Proposition 4.2 the structure of R(α0,G,U) isknown, this allows us to understand the structure of R(αG,F).

Proposition 6.1.

(1) Let x ∈ SL(n, R), 1 ≤ i 6= j ≤ n and x′ ∈ Uxij. Let y ∈ N x

0 .Then the correspondence

Px,yij : x′ 7→ Fy

ij ∩N x′

0

is a well defined and continuous map taking Uxij to Fy

ij.(2) Let x ∈ SL(n, R), 1 ≤ i 6= j ≤ n and x′ ∈ Fx

ij. Let y ∈ N x0 .

Then the correspondence

Px,yij : x′ 7→ Uy

ij ∩N x′

0

is a well defined and continuous map taking Fxij to Uy

ij.

Maps Px,yij and Px,y

ij are called canonical projections.

Proof: (1) We have x′ = eij(t) · x for some t ∈ R and y = d · x forsome d ∈ D+

We first show that the intersection of Fyij and N x′

0 exists whenevery and x′ are as above. It is immediate then, that such an intersectionis unique. It follows from the fact that αG is a conjugate of an actionwhich is C2-close to α0,G that the leaves of the corresponding Lyapunovfoliations are C0 close on compact sets. However the piece of the foli-ation Uij between x and x′ may be rather long. So in order to definea projection we first make it small. For that we choose an element ofthe action α0,G, namely an element in A which contracts Uij, i.e. suchthat χij(a) < 0 and sufficiently away from the Weyl chamber walls.Now consider the U -path along U z

ij between z := α0,G(a, y) = d(a, y) · y(where d(a, y) is some element in D+ since αG moves points along theneutral leaves only) and z′ := d(a, y)d · x′ = eij(t

′) · z. For the properlychosen a, the length of this piece of U z

ij leaf is sufficiently small so thatthe neutral leaf through z′ intersects the leaf F z

ij at a point w. Now we

define y′ := αG(a−1, w). From the construction this point lies both onFy

ij and N x′0 , so we may define Px,y

ij (x′) = y′

(2) Let x ∈ SL(n, R), let 1 ≤ i 6= j ≤ n and let x′ ∈ Fxij. Let

y ∈ N x0 , y = d · x for some d ∈ D+. Choose an element a ∈ A which

contracts Fij. Translate the leaf Fxij by this element and obtain points

z := αG(a, x) and z′ := αG(a, x′) ∈ F zij. Now if a was chosen properly,

the piece between z and z′ is small enough so that N z′0 intersects U z

ij

at some point w. Now since z = αG(a, x) ∈ N y0 we have that z = d1 · y


for some d1 ∈ D+. Therefore the point y′ := d−11 · w lies both on Uy

ij

and on N x′0 , so we may define Px,y

ij (x′) := y′. �

Remark. Canonical projections are mutually inverse:

Py,xij (Px,y

ij (x′)) = x′ and Py,xij (Px,y

ij (x′)) = x′

Definition 12. Let c : x1, ..., xm−1, xm = x1 be a U -cycle with initialpoint x1. Then the canonical projection of this path at a point y :=d · x1, d ∈ D+ is an F -path Px1,y(c) : y1 = y, y2, . . . , ym such that foreach k ∈ {1, ..,m− 1} we have

yk+1 := Pxk,yk

i(k)j(k)(xk+1).

The projected path Px1,y(c) need not be closed, but since the pro-jection is along the leaves of the neutral foliation N0 we have thatym ∈ N y1

0 i.e. ym = dx,y(c) · y1 for some dx,y(c) ∈ D+. The map(x, y, c) → dx,y(c) is continuous due to continuity of projections.

Similarly we define the reverse projection Px1,y(c) of an F -cycle cwith an initial point x1 and to a (possibly open) U -path starting at y,using canonical projections Px,y

ij . The difference between the end-points

along the leaves of N0 will be denoted by dx,y(c).

Lemma 6.2. Canonical projections have the following properties:

(1) Px,y : ASsU(α0,G)x → ASs

F(αG) andPx,y : ASs

F(αG)x → ASsU(α0,G).

(2) Px,y : ASrsU (α0,G)x → ASrs

F (αG) andPx,y : ASrs

F (αG)x → ASrsU (α0,G).

(3) Px,y : ASU(α0,G)x → ASF(αG) andPx,y : ASF(αG)x → ASU(α0,G).

(4) Canonical projections Px,y map α0,G-reducible U-cycles in Cx(U)to αG-reducible F-cycles in Cy(F) and conversely Px,y map αG-reducible F-cycles in Cx(F) to α0,G-reducible U-cycles in Cy(U).

Proof: This Lemma is an immediate consequence of the Definition12 of canonical projections i.e. the fact that any canonical image of astable leaf is a stable leaf, and the continuity of canonical projections.�

Remark. Since reducible cycles project to reducible cycles it followsimmediately that canonical projections of any two cycles in the samereducibility class, have the same end-points on the leaf of N0 i.e. maps


dx,x and dx,x do not depend on the representative from the given re-ducibility class.

6.3. Canonical projections of U-cycles.

Lemma 6.3.

(1) On the covering space, projection of every contractible U-cycleis an F-cycle.

(2) Projection of every U-cycle which is not contractible is an F-path such that the distance between its end-points is small of theorder of the smallness of the perturbation αG.

Proof. (1) The claim for contractible U -cycles is a consequence of thediscussion preceding Proposition 4.2 and of Lemma 6.2 4. Namely everycontractible U -cycle in SL(n, R) is represented by a relation in thegroup SL(n, R) which is α0,G-reducible due to the Steinberg theoremand Lemma 4.1. Now since α0,G-reducible U -cycles project to αG-reducible F -cycles, the claim follows.

(2) If a U -cycle u in SL(n, R) is not contractible then it is in thesame homotopy class, thus in the same reducibility class, as a cyclegenerated by the relation {−1,−1} = id. Such a cycle is given by therelation (e12(−1)e21(1)e12(−2)e21(1)e12(−1))2 = 1 so it is symmetric,so a U cycle u0 generated by this relation projects in the same way asits inverse. So even though u2

0 projects to a closed F path, the cycleu0 may project to an open F -path with opening d ∈ D+. It is stilltrue however that due to the smallness of the perturbation d must besmall for this u0 and thus for any other U -cycle u which is in the samereducibility class as u. �

6.4. Canonical projections of F-cycles. Let now Cx denote thecollection of contractible F -cycles starting at x in SL(n, R) and definethe following subset of D+:

D(x) := {dx,x(c) : c ∈ Cx}where dx,x(c) is defined in Section 6.2 via endpoints of U paths whichare obtained as canonical projections Px,x(c).

Since for c ∈ Cx the element dx,x(c) ∈ D+ does not depend on theinitial point of the projection, the set D(x) is a subgroup of D+ and wewill denote dx,x(c) simply by d(c) whenever it is clear what the initialpoint x is.

Lemma 6.4. For all x, D(x) is the same subgroup of D+ and wedenote it by D.


Proof. Let c be in Cx and such that d(c) 6= 1. Let z ∈ Fxij and consider

the closed path beginning at z which is obtained from c by adding atpoint x the piece of the leaf of Fij from z to x at the beginning and inthe opposite direction at the end of c. Call this new path c.

The endpoints of any two projections of a two point path contained ina leaf of some Fij are obtained by the action of the same element of thefull WCF. If d(c) = diag(d1, . . . , dn) then direct matrix multiplicationshows that the inclusion

eij(t) · d(c) · eij(sij(t)) ∈ D+

implies for every pair i, j, i 6= j

disij(t) = −djt.

Moreover, in this case we have:

eij(t) · d(c) · eij(sij(t)) = d(c),

so by moving along the leaves of foliations Fij the N0 distance be-tween the endpoints of any projection to U -paths does not change. Inparticular D(x) does not change along the leaves of foliations Fij.

By Lemma 6.1, foliations Fij, 1 ≤ i, j ≤ n constitute a transitivesystem on SL(n, R). This implies that D(x) is constant everywhere.

�

Lemma 6.5. The group D is discrete.

Proof. For any d ∈ D+ there is the following canonical way of repre-senting it as a product of elementary diagonal matrices hij(t). Namely,

d = h12(t1) · h23(t2) . . . hn−1,n(tn−1)

and for each i ∈ {1, . . . , n− 1} and any t ∈ R∗, hi,i+1(t) is defined as

hi,i+1(t) := ei,i+1(t)ei+1,i(−t−1)ei,i+1(t)ei,i+1(−1)ei+1,i(1)ei,i+1(−1).

This representation is unique. For any x ∈ SL(n, R) the aboveexpression for d defines a U -path which consists of 6(n− 1) arcs alongleaves of foliations Uij for various (i, j). We call such a path a standardU -path. So to any d ∈ D and to any point x there corresponds astandard U -path which we denote by ud,x which connects x and dx.

Now assume that the connected component of the identity D0 in thegroup D is nontrivial. Pick d ∈ D0 of norm one. Let c be an F -cyclein Cx such that d(c) = d. Now to d of norm one there correspondsa standard U -path ud,x consisting of a fixed number of arcs (at most6(n − 1) arcs) and moreover, the lengths of these arcs are uniformly


bounded since d is of norm 1. Let u be the closed U path which consistsof the standard U -path ud,x and the projection u = Px,x(c).

By Lemma 6.3 the projection Px,x(u) is a closed F -path if u is con-tractible, and since c is a closed F -path, it follows that the projectionof the standard path cd := Px,x(ud,x) is also a closed F -path at x.

Since the length of arcs in ud,x is uniformly bounded, each of the linksof Px,x(ud,x) is close (the order of the smallness of the perturbation)to the corresponding link of ud,x and for any d the number of links ofPx,x(ud,x) is at most 6(n− 1). Hence the end point dx is close to x i.e.d has to be small which is in contradiction with d being of norm 1.

If u is not contractible then one only needs to double it in order toget a contractible path, so the argument above applies in the same way,i.e. d is obtained by projecting a U cycle with fixed number of links ofbounded length, thus has to be small, which again contradicts d beingof norm 1.

Thus D0 = {id} and D is discrete. �

Corollary 6.2. Every F path of sufficiently small diameter is αG-reducible.

Lemma 6.6. The group D is trivial.

Proof: Without loss of generality (taking the double if necessary) ifD is not trivial we may suppose that there is a (very long) contractibleF -cycle c with base point x whose canonical projection is not closed.Let cs, 0 ≤ s ≤ 1 be a homotopy fixing x between c = c0 and thetrivial cycle c1.

For any ε > 0 one can find M such that we can construct a finitesequence of F cycles c0 = c, c1 . . . , cM = c1 such that for k = 0, 1 . . . , Mck is C0, ε close to ck/M as a parametrized path, i.e. a map [0, 1] → X.If M is chosen large enough them C0 distance between ck and ck+1 willbe less than 2ε for each k, 0 ≤ k ≤ M − 1.

If ε is chosen small enough this implies that the cycles ck and ck+1 areαG equivalent. For, the cycle ck+1∗(ck)−1 is conjugate to a compositionof cycles of diameter less than 4ε. Each of those cycles is αG-reducibleby Corollary 6.2. Hence the original cycle c is equivalent to the trivialcycle and hence is αG-reducible. This implies by Lemma 6.2 that itscanonical projection is a U -cycle, a contradiction. �

Corollary 6.3. Every projection of every F-cycle on the cover SL(n, R)is a U-cycle.

Proposition 6.2. For every contractible F cycle c in X, c or c ∗ c isαG-reducible.


Proof. Let c be a contractible F -cycle in X. Then it lifts to a con-tractible F -cycle c in SL(n, R) starting at some x. Since dx,x(c) = id,the U -path u = P x,x(c) is a U -cycle on the cover and therefore it isreducible, or reducible if doubled (Lemma 6.3), so by Lemma 6.2, thesame holds for c and thus the same holds for c. �

7. Proof of Theorems 2 and 3

7.1. Simple transitivity of F-holonomy: proof of Theorem 3.The F -holonomy group acts transitively due to the transitivity of thesystem of foliations Fij. Now let x, y be on the same leaf of N0 andconsider two different F -paths p1 and p2 connecting x and y. Thenthe F -path c′ corresponding to the cycle c := p1 ∗ p−1

2 starting at anyother point on the leaf of N0 is also closed. If c is αG-reducible this isdue to the fact that projections along the leaves of N0 from leaves of Ffoliations to leaves of F -foliations preserve reducibility classes, so c′ isalso reducible, so it is an F -cycle. If c is not in the trivial reducibilityclass then the map taking c to the distance between the endpoints of c′

at some different point on the leaf ofN0 gives a homomorphism from thegroup of αG-reducibility classes into D+ and every such homomorphismis trivial by Proposition 6.2 and the fact that there are no non-trivialhomomorphisms exist from Γ into D+ ([19, Theorem (4), Chapter 1]).

Hence the corresponding holonomy map is an identity and the holo-nomy maps corresponding to p1 and p2 are the same.

7.2. Cocycle rigidity for a C2-small perturbation of a genericrestriction: proof of Theorem 2. Let αG be a C2-small pertur-bation of a generic restriction α0,G. Let F denote the collection ofLyapunov foliations of αG. Let αG be the action obtained by conju-gating αG by the Hirsch-Pugh-Shub homeomorphism as described inSection 2.1.1. As before F denotes the collection of Lyapunov foliationsof αG.

By Proposition 6.2 the assumption 1 in Proposition 3.2 is satisfiedfor F -foliations of αG. Since this property is preserved by topologicalconjugacy, Lyapunov foliations for αG i.e. F -foliations, also satisfy thefirst assumption in Proposition 3.2. Now Theorem 2 is an immediateconsequence of Proposition 3.2 and the fact that due to the higher-rank assumption any homomorphism from π1(X) into Rl is trivial [19,Theorem (4), Chapter 1].


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Department of Mathematics, Harvard University, Cambridge MA02138

E-mail address: [email protected]

Department of Mathematics, The Pennsylvania State University,University Park, PA 16802

E-mail address: katok [email protected]: http://www.math.psu.edu/katok a

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