global rigidity of higher rank abelian anosov algebraic actions

44 Pages·2013·0.5 MB·English

GLOBAL RIGIDITY OF HIGHER RANK ABELIAN
ANOSOV ALGEBRAIC ACTIONS
FEDERICO RODRIGUEZ HERTZ AND ZHIREN WANG
Abstract. We show that all C∞ Anosov Zr-actions on tori and
nilmanifolds without rank-one factor actions are, up to C∞ conju-
gacy, actions by automorphisms.
Contents
1. Introduction 2
1.1. Main result 2
1.2. Background of the problem 2
1.3. Organization of paper 4
2. Preliminaries 5
2.1. General settings 5
2.2. Franks-Manning conjugacy 6
2.3. Coarse Lyapunov decomposition 7
2.4. Correspondence between foliations 9
2.5. Rank assumption and ergodicity of generic elements 10
3. Smooth conjugacy in certain coarse Lyapunov subspaces 12
3.1. The strong stable – weak stable – unstable decomposition 12
3.2. The cohomological equation 15
3.3. Smoothness of parameters 18
3.4. Solving the linearized equation 20
3.5. Establishing smoothness 23
4. Construction of new Anosov elements 25
4.1. Local description of the strong stable foliation 25
4.2. Coarse Lyapunov decomposition over an invariant measure 27
4.3. Smoothness of strong stable leaves 30
4.4. Crossing the Weyl chamber wall 35
5. Conclusion of the proof 36
Appendix A. A regularity theorem along H¨older foliations 37
References 42
1
2 F. RODRIGUEZ HERTZ AND Z. WANG
1. Introduction
1.1. Main result. Consider a Zr-action α on a torus, a nilmanifold or
an infranilmanold M by C∞ diffeomorphisms. The action by n ∈ Zr
is written as αn. Following results of Franks [Fra69] and Manning
[Man74] and using the commutativity of the action, one can check
easily that α is topologically conjugate to an action ρ : Zr (cid:121) M by
affine automorphisms. The action ρ is called the linearization of α.
Recall a compact nilmanifold is the quotient of a simply connected
nilpotentLiegroupGbyacocompactdiscretesubgroupΓ(see[Mal51]),
and a compact infranilmanifold is a manifold that is finitely covered
by a compact nilmanifold. A linear automorphism of G/Γ is a
homeomorphism that is the projection of some Γ-preserving automor-
phism of G. An affine automorphism of G/Γ is a homeomorphism
f : G/Γ (cid:55)→ G/Γ such that f(gx) = f (g)f(x) for all g ∈ G, x ∈ M,
0
where f is an automorphism of G. Equivalently, an affine automor-
0
phismofG/ΓisthecompositionofalinearautomorphismofG/Γanda
left translation. An affine automorphism of a compact infranilmanifold
is a homeomorphism that lifts to an affine nilmanifold automorphism
on a finite cover. We prove in this paper:
Theorem 1.1. Suppose α : Zr (cid:121) M is a C∞ action on a compact
infranilmanifold M where r ≥ 2, and let ρ be its linearization. Assume
that there exists n such that αn0 is an Anosov diffeomorphism, and
0
that ρ has no rank-one factor. Then α is conjugate to ρ by a C∞
diffeomorphism.
Arank-onefactoristheprojectionofρtoaquotientinfranilmanifold,
which is, up to finite index, generated by a single ρn. For details, see
§2.5.
Even in the case of tori, the theorem is new.
A priori, it may happen that M is equipped with an exotic smooth
structure ω instead of the standard structure ω inherited from the
0
covering Lie group G. α is smooth with respect to ω, and ρ is smooth
with respect to ω . In this case, the smooth conjugacy in the theorem
0
shouldbeunderstoodasadiffeomorphismbetween(M,ω)and(M,ω ).
0
This in particular shows (M,ω) is actually a standard infranilmanifold
and yields a contradiction. Therefore, the theorem implies:
Corollary 1.2. Exotic infranilmanifolds admit no Anosov Zr-action
without rank-one factor.
1.2. Background of the problem. For a Zr-action by diffeomor-
phisms on a compact manifold, there is a sharp contrast between the
r ≥ 2 case and the r = 1 case. The simplest example for a Z-action by
GLOBAL RIGIDITY OF HIGHER RANK ANOSOV ALGEBRAIC ACTIONS 3
a single Anosov diffeomorphism is a hyperbolic toral automorphism A.
In this case, one can modify A near a fixed point to get a new diffeo-
morphism f of the torus which is close to A in the C∞-topology, but
cannot be conjugate to A by a diffeomorphism because the eigenvalues
of its derivatives differ from those of A.
On the other hand, higher rank abelian Anosov actions enjoy more
rigidity in this aspect. An action is said to be C∞-locally rigid if all C1-
perturbations are C∞-conjugate to the original action. Local rigidity
was first proved for Cartan actions on tori by Katok and Lewis [KL91],
and later extended to some quite general classes of actions in the works
of Katok and Spatzier [KS94,KS97] and Einsiedler and Fisher [EF07].
Such phenomena motivated the following global rigidity conjecture by
Katok and Spatzier (see [Gor07]):
Conjecture 1.3. When r ≥ 2, all “irreducible” Anosov genuine Zr-
actions on compact manifolds are C∞-conjugate to actions on tori,
nilmanifolds or infranilmanifolds by affine automorphisms.
Some progresseshave been madetowards the conjecture in the works
of Kalinin-Spatzier [KS07b] and Kalinin-Sadovskaya [KS06,KS07a].
However, these results depend on strong conformality assumptions on
the restriction of the action to coarse Lyapunov foliations, as well as
other algebraic conditions which force the manifold to be a torus. And
the conjecture is widely open.
Attention has been focused on the following natural weaker form of
Conjecture 1.3:
Conjecture 1.4. When r ≥ 2, all “irreducible”1 Anosov genuine Zr-
actions on compact infranilmanifolds are C∞-conjugate to actions by
affine automorphisms.
Such global rigidity results have been obtained in two special cases,
with different extra assumptions that are fairly disjoint to each other.
In [RH07], the first author proved that Conjecture 1.4 holds for a
class of actions on tori, which in particular includes Cartan actions,
i.e. Zr-actions on Tr+1. The restrictions posed in [RH07] require the
rank r grows linearly in terms of the dimension of the torus.
In [FKS11,FKS13], Fisher, Kalinin and Spatzier established Conjec-
ture 1.4 under the additional assumption that all Weyl chambers of
ρ contains at least one Anosov element of α. Another technical hy-
pothesis, total non-symplecticity, was made in [FKS11] and was later
dropped in [FKS13]. They also raised the question of whether there are
1As [FKS13] and Theorem 1.1 shows, it turns out that irreducibility is not a
crucial issue here.
4 F. RODRIGUEZ HERTZ AND Z. WANG
Anosov genuine Zr-actions on exotic tori for r ≥ 2, based on examples
constructed by Farrel and Jones [FJ78] in the r = 1 case, and observed
that the existence of such actions would contradict Conjecture 1.4.
Theorem 1.1 confirms Conjecture 1.4, without assuming irreducibil-
ity. This leads to a negative answer, Corollary 1.2, to the question
about actions on exotic tori,
Itwasremarkedin[FKS13]that, sincethereexistnon-Anosovdiffeo-
morphisms that are Ho¨lder conjugate to Anosov toral automorphisms
(constructed by Gogolev in [Gog10]), a priori, generic elements in α
might not be Anosov. We will rule out this possibility, and hence ver-
ify that the hypothesis of [FKS13] always holds. The proof of the fact
that generic elements are Anosov will borrow ideas from both [RH07]
and [FKS13], while following a new scheme.
1.3. Organization of paper. Our strategy is to show that if a Weyl
chamber of the linearized action ρ contains an Anosov element, then
any adjacent Weyl chamber also contains Anosov elements. This will
show that there are Anosov elements in all the Weyl chambers, al-
lowing to apply Fisher, Kalinin and Spatzier’s theorem and establish
Conjecture 1.4.
In contrast to methods adopted in previous literatures (for example,
[MQ01], [FKS13]) when establishing smooth rigidity on nilmanifolds,
our approach directly deals with individual coarse Lyapunov foliations
on the nilmanifold, instead of going through an induction on the step
of nilpotency which starts with the underlying torus factor of the nil-
manifold.
In Section 2, we state preliminary facts regarding actions on nilman-
ifolds and the coarse Lyapunov foliation for the linearized action ρ.
The meaning of the assumption that ρ has no rank-one factor will be
explained.
WhenmovingfromanAnosovelementαn0 fromaWeylchambertoa
new element αn in a neighboring chamber, one crosses a Weyl chamber
wall that corresponds to a coarse Lyapunov subgroup V, which in turn
gives rise to a subfoliation that is contracted by the old element but
expanded by the new one under the linear action. With respect to the
automorphism ρn0, the group G can be decomposed into three parts:
the strong stable subgroup Gss, V and the unstable subgroup Gu. In
Section 3, we locally parametrize the Franks-Manning conjugacy H,
and project it into V in the GssVGu decomposition to get a coordinate
function. Then we verify that this coordinate function is smooth along
the stable foliation Ws of αn0 by solving a non-linear cohomological
α
equation inside the Lie algebra of V.
GLOBAL RIGIDITY OF HIGHER RANK ANOSOV ALGEBRAIC ACTIONS 5
In Section 4, it is shown that level sets of the coordinate function
within αn0-stable leaves forms a subfoliation which is contracted by αn.
In order to show that this new subfoliation has smooth leaves, we check
thatthedifferentialofthisfunctionalongWs doesnotdegenerate. This
α
is achieved by applying Pesin theory to an α-invariant measure ν that
is supported on the points where this condition is not satisfied. More
precisely, we will rely on facts from [LY85] on the almost everywhere
Lipschitz continuity of the strongly stable foliation inside the stable
leaves, to show that the degeneracy of the differential would cause the
restriction of H to the stable leaves to have a zero Jacobian, which
will be proved to be impossible. Eventually, the existence of the new
subfoliation will allow us to employ the quasi-Anosov criterion of Man˜´e
[Man˜77] to show that αn is also Anosov. The final lines of the proof
are included in Section 5.
The proof of the smoothness of h in Section 3 relies on a result stat-
V
ing that if all partial derivatives of a Ho¨lder function along a H¨older fo-
liation with smooth leaves belong to the distribution space over Ho¨lder
functions, then the function is smooth along the foliation. This is a
strengthening of a theorem of Rauch and Taylor [RT05] and the proof
is included in Appendix A.
Acknowledgments. FedericoRodriguezHertzwassupportedbyNSF
grant DMS-1201326. Zhiren Wang was supported by NSF grant DMS-
1201453 and an AMS-Simons travel grant.
2. Preliminaries
2.1. General settings. In this paper α will be a C∞-action by Zr on
a compact nilmanifold (or more generally, a compact infranilmanifold)
M, that is, a group morphism α : n (cid:55)→ αn from Zr to the group
Diff∞(M) of C∞ diffeomorphisms of M.
We will always assume that there exists at least one n for which αn0
0
is Anosov, and that the linearization action ρ : Zr (cid:121) M, which will be
defined below in §2.2, has no non-trivial rank-1 factor (see Definition
2.8).
Throughout the paper except in §1.2, it will be assumed that the dif-
ferential structure of M is the one coming from the standard structure
of the covering nilpotent Lie group G. Let N denote the dimension
dimM = dimG.
Recall that a diffeomorphism f on a Riemannian manifold M is
Anosov if there is a continuous splitting of the tangent bundle TM =
Es ⊕ Eu such that Df preserves the splitting and for some C,λ > 0
f f
6 F. RODRIGUEZ HERTZ AND Z. WANG
and every k ≥ 0,
(cid:107)Dfk(v)(cid:107) ≤ Ce−λk(cid:107)v(cid:107), ∀v ∈ Es;
f
(cid:107)Df−k(v)(cid:107) ≤ Ce−λk(cid:107)v(cid:107), ∀v ∈ Eu.
f
In this case M is everywhere foliated by the unstable foliation Wu,
f
as well as by the stable foliation Ws, which are respectively tangent
f
to Eu and Es. The leaves of Wu and Ws are given by
f f f f
(cid:0) (cid:1)
Wu(x) ={y ∈ Y| lim dist f−n(x),f−n(y) = 0};
f
n→∞
(2.1)
(cid:0) (cid:1)
Ws(x) ={y ∈ Y| lim dist fn(x),fn(y) = 0},
f
n→∞
and are locally immersed C∞ submanifolds. But in general these foli-
ations are not smooth in the sense that the distribution Eu (resp. Es)
f f
does not necessarily vary in a C∞ way along the transversal direction
to Wu (resp. to Ws).
f f
We will write in the rest of the paper Eu, Es, Wu and Ws respec-
α α α α
tively for Eu , Es , Wu and Ws .
αn0 αn0 αn0 αn0
As α is a commutative action, it is easy to check from (2.1) that the
foliations Ws and Wu are invariant under αn for all n ∈ Zr.
α α
2.2. Franks-Manning conjugacy. As αn0 is an Anosov diffeomor-
phism of a compact nilmanifold M = G/Γ, by the classical works of
Franks [Fra69] and Manning [Man74], αn0 is conjugate to an Anosov
affine automorphism by a bi-H¨older homeomorphism H : M (cid:55)→ M
which is homotopic to identity. By [Wal70] any diffeomorphism of M
commuting with an affine automorphism is an affine automorphism,
and thus H conjugates the entire action α to an action ρ : Zr (cid:121) M by
affine automorphisms. To be precise,
αn = H−1 ◦ρn ◦H,∀n ∈ Zr. (2.2)
By [FKS11, Prop. 2.4], the image µ := H−1(Leb) of the Lebesgue
∗
measureonTN istheuniqueα-invariantabsolutelycontinuousmeasure
on TN. Moreover, its density function is positive and C∞.
In fact, for each n ∈ Zr, αn induces an action on the fundamental
group Γ. Every automorphism of Γ naturally extends to a Γ-preserving
automorphism of G, and hence induces an linear automorphism of the
nilmanifold M = G/Γ. Hence α induces an action on M by linear
automorphisms. which is exactly the linear part of ρn.
By abusing notation, we will denote by ρ as well the linear part
of the lift of the action ρ to G. That is, for all g ∈ G, x ∈ M,
ρn(gx) = (ρng)(ρnx)
In addition, the differential of ρn at the identity gives a Zr-action
on the Lie algebra g of G. Abusing notation again, we use ρ to call
GLOBAL RIGIDITY OF HIGHER RANK ANOSOV ALGEBRAIC ACTIONS 7
this action as well. In particular, we have the relation exp(ρnv) =
ρnexpv,∀v ∈ g.
As G is simply connected and nilpotent, the exponential map exp is
a diffeomorphism between g and G, with inverse log : G (cid:55)→ g.
2.3. Coarse Lyapunov decomposition. We now introduce the Lya-
punov decomposition of the Lie algebra.
Definition 2.1. Given a Zr-action ρ on G by automorphisms and a
linear functional χ ∈ (Rr)∗. The Lyapunov subspace corresponding
to χ is
(cid:12) (cid:12)
log(cid:12)ρnv(cid:12)−χ(n)
vχ = {v ∈ g\{0} : lim = 0,∀n ∈ Zr}∪{0}.
|n|→∞ |n|
χ is called the Lyapunov exponent of vχ.
Lemma 2.2. vχ is indeed a vector subspace of g. There are only finitely
many non-trivial vχ’s and
(cid:77)
g = vχ (2.3)
χ
Proof. Fix a basis e ,··· ,e of Zr. Then g⊗ C is the direct sum of
1 r R
generalized eigenspaces ker(ρe1 − λI)dimg. Since ρ is a commutative
action, each of these generalized eigenspaces are invariant under every
ρei. Hence each generalized eigenspaces splits into the direct sum of
smaller generalized eigenspaces of ρe2, which are again invariant un-
der all the ρei’s. Keeping doing this, one can eventually decompose
the entire vector space g⊗ C as the direct sum of finitely many sub-
R
(cid:76)
spaces W , each of which is a common generalized eigenspace of
j j
ρe1,··· ,ρer. On every W , the commuting matrices ρe1,··· ,ρer can
j
be upper triangularized together, and each ρei has only one eigenvalue
ζ . Actually ρei| can be written as ζ A where A is an upper
ij Wj ij ij ij
triangular matrix whose diagonal entries are equal to 1. It follows that
for all n = (cid:80)r n e in Zr, W is also a generalized eigenspace of ρn
i=1 i i j
with a unique eigenvalue ζn = (cid:81)r ζni. And ρn| = ζnAn where
An = (cid:81)r Ani. One can eajsily chei=ck1 nij(cid:55)→ ζn and nWj(cid:55)→ Anj arje group
j i=1 ij j j
morphisms from Zr, respectively to C× and to End(W ).
j
Furthermore, entries of An are polynomial in n. Hence, for any
j
non-zero v ∈ W , ρnv has polynomial entries as n varies. Therefore a
j ζn
j
non-zero vector v ∈ g (or g⊗ C) satisfies the asymptotic formula in
R
Definition 2.1 if and only if it lies in the direct sum of the g ∩ W ’s
j
such that the linear map n (cid:55)→ log|ζn| is equal to χ. This direct sum is
j
8 F. RODRIGUEZ HERTZ AND Z. WANG
exact vχ. Moreover, g = (cid:76) (g∩W ) = (cid:76) vχ, and thus the number
j j χ
of non-trivial vχ’s is finite. (cid:3)
Lemma 2.3. For all Lyapunov subspaces vχ1 and vχ2, the relation
[vχ1,vχ2] ⊂ vχ1+χ2 holds.
In particular, if the Lyapunov exponent χ + χ is absent in the
1 2
Lyapunov decomposition, then vχ1 and vχ2 commute.
Proof. Suppose v ∈ vχ1, w ∈ vχ2, we wish to show that [v,w] ∈ vχ1+χ2.
By decomposing into components if necessary , we can assume v and
1
v are respectively in common generalized eigenspaces W and W con-
2 1 2
structed as in the proof of Lemma 2.2, with respective eigenvalues ζn
1
and ζn. Then
2
ρn[v,w] (cid:2)ρnv,ρnw(cid:3) (cid:20)ρnv ρnw(cid:21)
= = , (2.4)
ζnζn ζnζn ζn ζn
1 2 1 2 1 2
haspolynomialentriesintermofn. But|ζn| = eχj(n) forj = 1,2. Thus
j
(cid:12) (cid:12)
log(cid:12)ρn[v,w](cid:12)−χ(n)
thevector[v,w], ifitdoesn’tvanish, satisfieslim =
|n|→∞ |n|
0 with χ = χ +χ . This completes the proof. (cid:3)
1 2
Definition 2.4. For a Lyapunov exponent χ that is present in the Lya-
punov decomposition, the corresponding coarse Lyapunov subspace
is
(cid:77)
v[χ] = vχ(cid:48).
χ(cid:48)=cχ,c>0
So a coarse Lyapunov subspace is a direct sum of all Lyapunov sub-
spaceswhoseexponentsarepositivelyproportionalin(Rr)∗. Whentwo
vectors belong to different coarse Lyapunov subspaces, one can find an
n such that ρin expands one of the vectors but contracts the other as
i → ∞. However vectors from the same coarse Lyapunov subspace
cannot be distinguished in this way.
This gives the coarse Lyapunov decomposition
(cid:77)
g = v[χ] (2.5)
Both the Lyapunov decomposition (2.2) and the coarse Lyapunov
decomposition (2.5) are ρ-invariant.
One consequence Lemma 2.3 is that, as the Lyapunov exponents
in each coarse Lyapunov subspace v[χ] are positively proportional to
each other and hence closed under addition, v[χ] is a Lie subalgebra.
Therefore we have:
GLOBAL RIGIDITY OF HIGHER RANK ANOSOV ALGEBRAIC ACTIONS 9
Definition 2.5. To each coarse Lyapunov subspace v[χ] is associated
a closed connected subgroup V[χ] = expv[χ] ⊂ G, which will be called a
coarse Lyapunov subgroup.
Fixing n ∈ Zr, the direct sum of the vχ’s with χ(n) < 0 (resp.
> 0) is the stable (resp. neutral, unstable) Lie subalgebra of ρn and
is denoted by gs (resp. gu ). The corresponding Lie subgroups are
ρn ρn
denoted respectively by Gs and Gu .
ρn ρn
EachcoarseLyapunovsubspacev[χ] givesrisetotheWeyl chamber
wallkerχ ⊂ Rr, whichisahyperplanethatdividesRr intothepositive
(resp. negative) Lyapunov half space
L[χ],+ = {w ∈ Rr|χ(w) > 0} (resp. L[χ],− = {w ∈ Rr|χ(w) < 0}).
(2.6)
Remark 2.6. The definitions of kerχ, L[χ],+ and L[χ],− clearly do not
depend on the choice of χ representing [χ], and two different coarse
Lyapunov subspaces v[χ] and v[χ(cid:48)] give rise to the same Weyl chamber
wall if and only if the exponents χ and χ(cid:48) are negatively proportional,
in which case L[χ],+ = L[χ(cid:48)],− and L[χ],− = L[χ(cid:48)],+.
Definition 2.7. The connected components of Rr\(cid:83) kerχ are called
[χ]
Weyl chambers of ρ.
By construction, Weyl chambers are the minimal non-trivial inter-
sections of different Lyapunov half spaces.
2.4. Correspondence between foliations. As earlier, α is a Zr-
action on M, with Anosov element αn and linearization ρ. For sim-
0
plicity, let gs, gu, Gs and Gu denote respectively gs , gu , Gs and
ρn0 ρn0 ρn0
Gu . It is easy to check that for x ∈ M, Gsx, Gux are just the stable
ρn0
and unstable foliations Ws(x) and Wu(x) for the affine automorphism
ρ ρ
ρn0. Since α and ρ are conjugate by H, we have:
H(Ws(x)) = GsH(x), H(Wu(x)) = GuH(x). (2.7)
α α
In particular, for (cid:3) ∈ {s,u}, G(cid:3)x is a manifold with dimension
dimW(cid:3) andthusdimg(cid:3) = dimW(cid:3) = dimE(cid:3). Itfollowsthatdimgs+
α α α
dimgu = N = dimg. Since gs and gu have trivial intersection by con-
struction, g = gs ⊕ gu. In other words, ρn0 is a hyperbolic automor-
phism, or equivalently, for all Lyapunov exponent χ for ρ, χ(n ) (cid:54)= 0.
0
This implies there is no trivial exponent in the Lyapunov decompo-
sition (2.3) and justifies Definition 2.7.
10 F. RODRIGUEZ HERTZ AND Z. WANG
2.5. Rank assumption and ergodicity of generic elements. We
will assume ρ : Zr (cid:121) M has no rank-one factor in the sense below
following previous literatures, e.g. [KK01,KKS02].
Let M = G/Γ be a nilmanifold. Suppose there is a surjective mor-
˘ ˘
phism π : G (cid:55)→ G to a nontrivial nilpotent Lie group G of lower dimen-
sion, such that Γ is projected to another cocompact discrete subgroup
˘ ˘
Γ ⊂ G. In this case π defines a projection, which is still denoted by π,
˘ ˘ ˘ ˘
from M to M = G/Γ. We say the new nilmanifold M is an algebraic
factor of M.
Furthermore, suppose ρ is a Zr-action on M by affine automor-
phisms. If the linear part of ρ, which is an action by group auto-
morphisms of G and is denoted also by ρ, preserves the kernel of the
˘
projection π, then ρ descends to an action on M by automorphisms,
which is a factor action of ρ.
Definition 2.8. A Zr-action ρ on a nilmanifold M by automorphisms
is said to have a rank-one factor if ρ admits a factor action ρ˘ on a
˘
nontrivial algebraic factor M of M, and for some finite-index subgroup
Λ ⊂ Zr, the linearization {(ρ˘)n : n ∈ Λ} consists of a cyclic group of
affine automorphisms.
A k-step nilmanifold M = G/Γ arises as principal bundles. The
centerG(cid:48) = [G,G]isa(k−1)-stepnilpotentnormalsubgroupofG, and
(cid:0) (cid:1)
both M(cid:48) = G(cid:48)/(G(cid:48) ∩Γ) and M = (G/G(cid:48))/ Γ/(G(cid:48) ∩Γ) are compact.
0
M is a bundle over M with M(cid:48) fibres. M(cid:48) is a (k−1)-step nilmanifold.
0
M is a torus as G/G(cid:48) is abelian, and is naturally an algebraic factor
0
of M, called the maximal torus factor of M. Since G(cid:48) is proper in G,
dimM > 0 as long as dimM > 0.
0
Since automorphisms of G preserve the center subgroup G(cid:48), any ac-
tion ρ by automorphisms projects to M .
0
By [Sta99], having no non-trivial factor is actually equivalent to the
assumptionin[FKS13]thatthereisaZ2-subactionwhoseallnon-trivial
elements act ergodically. In fact, we have:
Lemma 2.9. Let ρ : Zr (cid:121) M be an action by nilmanifold automor-
phisms. Then the following are equivalent:
(1) ρ has no rank-one factor;
(2) There is a subgroup Σ ⊂ Zr such that Σ ∼= Z2 and ρn is an
ergodic toral automorphism for all n ∈ Σ\{0}.
(3) There are finitely many subgroups A ,··· ,A ⊂ Zr, all of rank
1 k
at most r −2, whose union contains every n ∈ Zr that fails to
act ergodically by ρ;