Local Rigidity for Cocycles

a r

X i

v

:m

a

t h /

3

3

2

3

6

v

3

[

m

a

t h

.

D

S

]

1

2

N

o

v

2

3

LOCAL RIGIDITY FOR COCYCLES DAVID FISHER AND G.A.MARGULIS Abstract.In this paper we study perturbations of constant co-cycles for actions of higher rank semi-simple algebraic groups and their lattices.Roughly speaking,for ergodic actions,Zimmer’s co-cycle superrigidity theorems implies that the perturbed cocycle is measurably conjugate to a constant cocycle modulo a compact val-ued cocycle.The main point of this article is to see that a cocycle which is a continuous perturbation of a constant cocycle is actu-ally continuously conjugate back to the original constant cocycle modulo a cocycle that is continuous and “small”.We give some applications to perturbations of standard actions of higher rank semisimple Lie groups and their lattices.Some of the results proven here are used in our proof of local rigidity for a?ne and quasi-a?ne actions of these groups.We also improve and extend the statements and proofs of Zim-mer’s cocycle superrigidity.1.Introduction Let G be a connected semisimple Lie group with no compact fac-tors and all simple factors of real rank at least two.Further assume G is simply connected as a Lie group or simply connected as an alge-braic group.The latter means that G =G (R )where G is a simply connected semisimple R -algebraic group.Let Γ

Γis measurably conjugate to a constant cocycle,modulo some com-pact noise.(See below for a precise formulation.)This theorem has many consequences for the dynamics of smooth actions of these groups.Even stronger results would follow if one could produce a continuous or smooth conjugacy.The main purpose of this paper is to prove that a perturbation of a constant cocycle is conjugate back to the constant

2 D.FISHER AND G.A.MARGULIS

cocycle via a small(and often continuous)conjugacy,modulo“small”noise.We also prove stronger and more general versions of the cocycle

superrigidity theorems than had previously been known.In particular, we do not need to pass to a?nite ergodic extension of the action and we obtain more general statements when k is non-Archimedean. Throughout we work with a more general group G.We let I be a?-nite index set and for each i∈I,we let k i be a local?eld of characteristic

zero and G i be a connected simply connected semisimple algebraic k i-group.We?rst de?ne groups G i,and then let G= i∈I G i.If k i is non-Archimedean,G i=G i(k i)the k i-points of G i.If k i is Archimedean, then G i is either G i(k i)or its topological universal cover.(This makes

sense,since when G i is simply connected and k i is Archimedean,G i(k i) is topologically connected.)Throughout the introduction,we assume that the k i-rank of any simple factor of any G i is at least two.

We?rst state a version of our main result for G actions and cocycles. Theorem1.1.Let G be as above,L=L(k)where L is an algebraic k-group and k is a local?eld of characteristic zero and letπ0:G→L be a continuous homomorphism.Let(S,μ)be a standard probability

measure space,ρa measure preserving action of G on S,andαπ

0:

G×S→L be the constant cocycle over the actionρgiven byαπ

(g,x)=π0(g).Assumeα:G×S→L is a Borel cocycle over the actionρsuch

thatαis L∞close toαπ

0.Then there exists a measurable mapφ:S→L,

and a cocycle z:G×S→Z where Z=Z L(π0(G)),the centralizer in L ofπ0(G),such that

(1)we haveα(g,x)=φ(gx)?1π0(g)z(g,x)φ(x);

(2)φ:S→L is small in L∞;

(3)the cocycle z is L∞close to the trivial cocycle

(4)the cocycle z is measurably conjugate to a cocycle taking values

in a compact subgroup C of Z where C is contained in a small

neighborhood of the identity.

Furthermore if S is a locally compact topological space,μis a Borel measure on S with supp(μ)=S andαandρare continuous then both φand z can be chosen to be continuous.

Remark:If k is Archimedean,point(4)implies that z is measurably conjugate to the trivial cocycle.

Before stating the analogous theorem forΓactions and cocycles, we need to recall a consequence of the superrigidity theorems[M1, M2,M3].We will use the notation introduced here in the statements below.If G is as above andΓ

LOCAL RIGIDITY FOR COCYCLES3πE:G→L and a homomorphismπK:Γ→L with bounded image such thatπ(γ)=πE(γ)πK(γ)andπE(Γ)commutes withπK(Γ).The su-perrigidity theorems imply that any continuous homomorphism ofΓinto an algebraic group is superrigid.This can be deduced easily from Lemma VII.5.1and Theorems VII.5.15and VII.6.16of[M3]. Theorem1.2.LetΓbe as above,L=L(k)be as in Theorem1.1and π0:Γ→L be a continuous homomorphism.Let(S,μ)be a standard probability measure space,ρbe a measure preserving action ofΓon S,

and letαπ

0:Γ×S→L be the constant cocycle over the actionρgiven

byαπ

(γ,x)=π0(γ).Assumeα:Γ×S→L is a Borel cocycle over the

actionρsuch thatαis L∞close toαπ

0.Then there exists a measurable

mapφ:S→L,and a cocycle z:Γ×S→Z where Z=Z L(πE0(G))such that

(1)we haveα(γ,x)=φ(γx)?1πE0(γ)z(γ,x)φ(x);

(2)φ:S→L is small in L∞;

(3)the cocycle z is L∞close to the constant cocycle de?ned byπK0

(4)z is measurably conjugate to a cocycle taking values in a compact

subgroup C of Z where C is contained in a small neighborhood

ofπK0(Γ).

Furthermore if S is a locally compact topological space,μis a Borel measure on S with supp(μ)=S andαandρare continuous then both φand z can be chosen to be continuous.

Remark:If k is Archimedean,point(4)implies that z is measurably conjugate to a cocycle taking values in the closure ofπK0(Γ).

To prove Theorems1.1and1.2,we prove a very general result about perturbations of cocycles over measure preserving actions of groups with property T.The result shows that any perturbation of a cocycle taking values in a compact group also takes values in a compact group, see Theorem5.4.

Use of(an extension and modi?cation of)Zimmer’s cocycle super-rigidity theorems is a key step in the proof of Theorem’s1.1and1.2. The cocycle superrigidity theorems are generalizations of the second author’s superrigidity theorems.Our strongest results require an inte-grability condition on the cocycles considered.

De?nition1.3.Let D be a locally compact group,(S,μ)a standard probability measure space on which D acts preservingμand L be a normed topological group.We call a cocycleα:D×S→L over the D action D-integrable if for any compact subset M?D,the function Q M,α(x)=supp m∈M ln+ α(m,x) is in L1(S).

4 D.FISHER AND G.A.MARGULIS

Any continuous cocycle over a continuous action on a compact topo-logical space is automatically D-integrable.We remark that a cocycle over a cyclic group action is D-integrable if and only if ln+ (α(±1,x) is in L1(S).

Theorem1.4.Let G,S,μ,L be as in Theorem1.1.Assume G acts ergodically on S preservingμ.Letα:G×S→L be a G-integrable Borel cocycle.Thenαis cohomologous to a cocycleβwhereβ(g,x)=π(g)c(g,x).Hereπ:G→L is a continuous homomorphism and c: G×S→C is a cocycle taking values in a compact group centralizing π(G).

Theorem1.5.Let G,Γ,S,L andμbe as Theorem1.2.AssumeΓacts ergodically on S preservingμ.Assumeα:Γ×S→L is aΓ-integrable, Borel cocycle.Thenαis cohomologous to a cocycleβwhereβ(γ,x)=π(γ)c(γ,x).Hereπ:G→L is a continuous homomorphism of G and c:Γ×X→C is a cocycle taking values in a compact group centralizing π(G).

The principal improvements over earlier results is that we do not need to pass to a?nite ergodic extension of the action and that we consider the case where k is a non-Archimedean?elds of characteristic0.This builds on work of the second author,Zimmer,Stuck,Lewis,Lifschitz, Venkataramana and others[L,Li,M1,M2,M3,Z2,Z1,Z4,Stu,V]. In the case where S is a single point,Theorem1.5is equivalent to the fact that all homomorphisms ofΓto algebraic groups are superrigid. Theorem1.4is equivalent to the same fact when applied to S=G/Γ. Remark:When k is non-Archimedean,it is not always the case that the algebraic hull of the cocycle is reductive unlike the case k=R treated in[Z4].

Remark:We also prove a result showing uniqueness of the homomor-phismπoccurring in Theorems1.4and1.5.See subsection3.8for details.

Remark:Most of the results here should be true for k i and k of positive characteristic as well,though additional arguments,similar to those in[V,Li]are apparently required.Some partial results in this direction are in[Li].

The main applications of our results on perturbations of constant cocycles are to studying perturbations of a?ne actions of G andΓ. De?nition1.6.1Let A and D be topological groups,and B

LOCAL RIGIDITY FOR COCYCLES5 2Let A and B be as above.Let C and D be two commuting groups of a?ne di?eomorphisms of A/B,with C compact.We call the action of D on C\A/B a generalized a?ne action.

3Let A,B,D andρbe as in1above.Let M be a compact Riemannian manifold andι:D×A/B→Isom(M)a C1cocycle.We call the result-ing skew product D action on A/B×M a quasi-a?ne action.If C and D are as in2,andα:D×C\A/B→Isom(M)is a C1cocycle,then we call the resulting skew product D action on C\A/B×M a generalized quasi-a?ne action.

Our notion of generalized a?ne action is from[F].The main ap-plication of our results on local rigidity of constant cocycles is as part of our work on local rigidity of volume preserving quasi-a?ne actions of G andΓon compact manifolds.We believe that volume preserving generalized quasi-a?ne actions on compact manifolds are locally rigid as well.As evidence for this,we have the following local entropy rigid-ity result.For any measure preserving actionρof D,we denote by hρ(d)the entropy ofρ(d).Let H be an algebraic group de?ned over R. We will refer to the connected component of the identity in H(R)as a connected real algebraic group.

Corollary1.7.Let H be a connected real algebraic group,Λ

This result generalizes the one in[QZ].Given the description of generalized standard a?ne actions below,the proof in[QZ]actually applies.We will prove Corollary1.7as a corollary of(part of)the proof of Theorems1.1and1.2.

We note here that our techniques prove local rigidity results for per-turbations of more general cocycles over actions of G andΓthan those in Theorems1.1and1.2.We can prove an analogous theorem for per-turbations of cocycles that are products of compact valued cocycles with constant cocycles.More generally,the original cocycle and the perturbed cocycle need not be cocycles over the same action,but only over actions that are“close”.For example if S is a topological space, then the actions being C0close is su?cient.(Since constant cocycles are cocycles over any action,one need only consider a single action in the formulations of Theorems1.1and Theorem1.2.)The proof of Corollary1.7then implies a local entropy rigidity result for generalized quasi-a?ne actions of G andΓ.The interested reader is welcome to

6 D.FISHER AND G.A.MARGULIS

adjust the proofs below to cover these situations,but for the sake of clarity we have restricted to the generality that we need for our next set of applications.

We now state a theorem which is used in our work on local rigid-ity of quasi-a?ne actions[FM1,FM2].This theorem shows that any perturbation of any quasi-a?ne action is continuously semi-conjugate back to the original action,at least“along hyperbolic directions”. Let H be a connected real algebraic group andΛ

Theorem1.8.Let H/Λ×M,ρ,D,Z andˉρbe as in the preceding para-graph.Given any actionρ′su?ciently C1close toρ,there is a con-tinuous D×Λequivariant map f:(H×M,ρ′)→(Z\H,ˉρ),and f is C0 close to the natural projection map.

For actions by left translations this follows from Theorems1.1and1.2. To prove Theorem1.8as stated here,we need a stronger result which is Theorem5.1in section5.Theorem1.8holds more generally for any skew product action of D on H/Λwhich is a?ne on H/Λand given by a cocycleι:D×H/Λ→Di?1ω(M)whereωis a volume form on M and M is compact.The version stated here is what is needed in[FM1].We note that,by Theorems6.4and6.5below any quasi-a?ne D action on H/Λ×M lifts to H×M on a?nite index subgroup D′

For example,for Sp(1,n),F?20

4and their lattices,our techniques can

be combined with the results of[CZ]to obtain local rigidity theorems for certain perturbations of certain cocycles of these groups.If variants

of Theorems1.4and1.5hold for Sp(1,n)and F?20

4and their lattices,

then Theorems1.1,1.2and5.1hold for these groups as well.

In section2we collect various standard de?nitions used throughout the paper.Section3concerns superrigidity for cocycles.Section4 proves that certain orbits in representation varieties are closed.Section 5contains the proof of our main results.The?nal section of the paper

LOCAL RIGIDITY FOR COCYCLES7 contains the proofs of Corollary1.7and Theorem1.8.This section also contains a detailed description of all a?ne actions of G andΓas above.

2.Preliminaries

We now collect various de?nitions that will be used in the course of the paper.

2.1.Algebraic groups.In this paper the words“algebraic group”mean a linear algebraic group de?ned over a local?eld k in the sense of[B2].Unless otherwise noted,throughout this paper k will be a local?eld of characteristic zero.For background on algebraic groups particularly relevant to what follows,we refer the reader to[M3,I.1-2].

2.2.Cocycles and ergodic theory.Given a group D,a space X and an actionρ:D×X→X,we de?ne a cocycle over the action as follows.Let L be a group,the cocycle is a mapα:D×X→L such thatα(g1g2,x)=α(g1,g2x)α(g2,x)for all g1,g2∈D and all x∈X.The regularity of the cocycle is the regularity of the mapα.If the cocycle is measurable,we only insist on the equation holding almost everywhere in X.Note that the cocycle equation is exactly what is necessary to de?ne a skew product action of D on X×L or more generally an action of D on X×Y by d(x,y)=(dx,α(d,x)y)where Y is any space with an L action.

We say two cocyclesαandβare cohomologous if there is a mapφ: X→L such thatα(d,x)=φ(dx)?1β(d,x)φ(x).Again we can de?ne the cohomology relation in any category,depending on how much regularity we seek or can obtain onφ.A cocycle is called constant if it does not depend on x,i.e.απ(d,x)=π(d)for all x∈X and d∈D.One can easily check from the cocycle equation that this forces the map πto be a homomorphismπ:D→L.Whenαis cohomologous to a constant cocycleαπwe will often say thatαis cohomologous to the homomorphismπ.The cocycle superrigidity theorems imply that many cocycles are cohomologous to constant cocycles,at least in the measurable category.

A measurable cocycleα:D×S→L is called strict if it is de?ned for all points in D×S and the cocycle equation holds everywhere instead of almost everywhere.For a dictionary translating facts about strict cocycles on homogeneous D-spaces to facts about homomorphisms of subgroups of D,see[Z2,Section4.2].

An action of a group D on a topological space X is called tame if the quotient space D\X is T0,i.e.if for any two points in D\X,there is an open set around one of them not containing the other.

8 D.FISHER AND G.A.MARGULIS

Given a locally compact group D and a discrete subgroupΓ

Let D be a compactly generated group,with compact generating set K.Let A be a metrizable,locally compact group and?x a distance function d:A×A→R.Given two measurable cocyclesα,β:D×S→A into a locally compact group A,we can de?ne a measurable function on S by d(α(d,x),β(d,x)).We say thatαandβare L∞close if there exists a smallε>0such that d(α(k,x),β(k,x)) ∞<εfor any k∈K. Let L be an algebraic k-group and L=L(k).Let a group D act ergodically on a measure space S and letα:D×S→L be a cocycle. There is a unique(up to conjugacy),minimal algebraic subgroup H in L such thatαis cohomologous to a cocycle taking values in H=H(k). The group H is referred to as the algebraic hull for the cocycle.This is a generalization the Zariski closure of a subgroup of an algebraic group. For more details,see chapter9of[Z2].

We recall that given any group D acting on a compact metric space X preserving a Borel measureμ,there is an ergodic decomposition ofμ. That is,there are Borel measuresμi on X,where eachμi is an invariant ergodic measure for the action of D,and the measureμis obtained as an integral of theμi over a speci?c measure?μon the space of measures on X.Furthermore,the measuresμi are mutually singular.

2.3.The space of actions.In the introduction,some statements are made about actions being C k close.Let D be a locally compact topo-logical group and X a smooth manifold.Since an action is a map D→Di?k(X)we can topologize the space of actions by taking the compact open topology on Hom(D,Di?k(X)).Two actions are C k close if they are close with respect to this topology.If D is compactly generated with compact generating set K,this means thatρandρ′are C k close if and only ifρ(d)?ρ′(d)?1is in a small neighborhood of the identity in Di?k(X)for all d∈K.Given a manifold or a space X equipped with an actionρ,we often write(X,ρ)to denote the space with the action.Similarly a map written(X,ρ)→(X′,ρ′)is a map of D-spaces or a D equivariant map.

LOCAL RIGIDITY FOR COCYCLES9

3.Superrigidity for Cocycles

In this section we prove Theorems1.4and1.5as well as some related

results.Our integrability condition allows us to use Oseledec’Multi-plicative Ergodic Theorem to obtain our general result.Some partial results below do not require the integrability condition.Theorem1.5 is deduced from Theorem1.4.The proof of Theorem1.4requires that

one?rst argue the case where L is semi-simple and then use the result in that case to prove the more general result.

Theorems1.4and1.5imply a general result on the algebraic hull of the cocycles considered.In fact,at least for G cocycles,this result is a

step in the proof of Theorem1.4,see Theorem3.10.It is proved in[M3] that for any?eld k and any homomorphismπ:Γ→L(k),the Zariski closure ofπ(Γ)is semisimple.This is equivalent to saying that the algebraic hull of the cocycleπ?β:G×G/Γ→L is semisimple.In[Z4], it is shown that if k=R,any G-integrable cocycleα:G×X→L has

algebraic hull reductive with compact center.If k is non-Archimedean, it is no longer the case that the algebraic hull is reductive.The following example shows that our results on the algebraic hull are sharp. Example3.1.We let J be a?nite index set and for each j∈J,we let k j be a local?eld of characteristic zero and H j be a connected sim-ply connected semisimple algebraic k j-group.We let H j=H j(k j)the

k j-points of H j and H= j∈J H j.We further assume that there is an irreducible latticeΛ

morphism and assume that Z H(π(G))contains a non-trivial unipotent subgroup U

The reader should note the following

(1)the above construction yields the same results when applied to

the restriction of the actions and cocycles to any latticeΓ

10 D.FISHER AND G.A.MARGULIS

(2)the construction gives non-trivial examples even when G=

G(R);

(3)one can take products of cocycles constructed as above with

constant cocycles to obtain cocycles whose algebraic hull is nei-

ther unipotent nor reductive;

(4)the argument above works for more general subgroups Z<

Z H(π(G))∩H l where K

group.One can construct examples where Z=F?U is a Levi

decomposition and the F action on U is non-trivial.

Let L be an algebraic group over k l and L=L(k l)and D=G orΓ. The above outline constructs cocyclesα:D×S→L of the formα=π·c whereπ:G→L is a continuous homomorphism and c:D×S→C is a cocycle taking values in a compact group C

We now brie?y indicate the plan of this section.Subsection3.1?xes notation for all of section3and contains some technical lemmas used throughout.In subsection3.2we prove a key technical result which shows that certain cocycles are cohomologous to constant co-cycles.Subsection3.3applies the results of subsection3.2to prove a variant of Theorem1.4where the algebraic hull of the cocycle is assumed to be semisimple.Subsection3.4proves some conditional re-sults on G-integrable cocycles,again using the results from subsection 3.2.We show how to use property T to control cocycles into amenable and reductive groups in subsection3.5and then prove Theorem1.4 in subsection3.6.Theorem1.5is also proven in subsection3.6mod-ulo some facts concerning G-integrability of certain induced cocycles. These facts are then proven in subsection3.7.Subsection3.8concerns the uniqueness of the homomorphismπin Theorems1.4and1.5.These results are used in subsection3.9to prove some results on cocycles with constrained projections.The result on cocycles with constrained pro-jections is required to prove Theorem5.1which is used in the proof of Theorem1.8.

3.1.Notations and reductions.In this subsection,we?x notations and de?nitions for all of section3.We also prove some technical lemmas that are used throughout this section.

The group G will be as speci?ed in the introduction,but we both weaken the rank assumption and make some preliminary reductions. Let S be the union of primes of Z and{∞}and let Q p be the p-adic completion of Q,where as usual,Q∞=R.By application of restriction

LOCAL RIGIDITY FOR COCYCLES11

of scalars,we can assume that each k i=Q p

α,where the p i are distinct

elements of the set S.As before,for the Archimedean factor,we can replace G i(R)by it’s topological universal cover.Actually this can

be done or not done for each simple factor independently,though we

simplify exposition by ignoring this nuance.Instead of assuming that

each simple factor of G i(k i)has k i-rank at least two,we let r i=k i-

rank(G i(k i))and de?ne the rank of G as i∈I r i and assume that the rank of G is at least two and that G has no non-trivial compact factors

(or,equivalently,that every simple factor of G i has k i-rank at least

one).

We specify a certain compact homogeneous G space,often called a boundary for G.Let P i

We?x(S,μ)to be a standard probability measure space.Also L

will denote an algebraic k-group and L=L(k).We denote by L0the

connected component of L and let L0=L0(k).As above,we apply

restriction of scalars and assume that k=Q p for some p∈S.

By a simple factor of G,we mean a subgroup F

De?nition3.2.Let(S,μ)be a?nite measure space.Given a group G acting ergodically on S preservingμ,we call the action weakly irre-ducible if for any rank one simple factor F

If no simple factor of G has rank1,this is equivalent to the ergod-

icity of the G action.This is weaker than the standard de?nition of irreducibility where it is assumed that all simple factors act ergodically [Z2].The de?nition of an irreducible action is motivated by properties of irreducible lattices.We call a latticeΓ

12 D.FISHER AND G.A.MARGULIS

a lattice is weakly irreducible if and only if the action of G on G/Γis weakly irreducible.

We will use the following elementary lemmas repeatedly.The?rst is obvious.

Lemma3.3.Let A be a group and letα:D×S→A andβ:D×S→A be cocycles over the action of a group D on a set S.Assume that β(D×X)is contained in a subgroup B

We letτi:G→G i(k i)be the natural projection.

Lemma3.4.Given a non-trivial continuous homomorphismπ:G→L there is i∈I such that k=k i and a k-rational homomorphismπi:

G i→L such thatπ=πi?τi.From this we can deduce:

(1)the Zariski closure ofπ(G)is semisimple and connected and;

(2)if L′→L is a k-isogeny,thenπlifts to a continuous homomor-

phismπ′:G→L′(k)

Proof.We?rst give the proof where all G i are k i-points of algebraic k i-groups and then describe the modi?cations necessary when G i is the topological universal of such a group.

Let the projection from G to G i beτi.Since k=Q p,by[M3,I.2.6] any continuous homomorphism of any G i into L is the restriction of rational map from G i to L.This implies there is an i and a rational homomorphismˉπ:G i→L such thatπis the restriction ofτi?ˉπ. Since G i is connected and semisimple and the characteristic of k is zero,it follows that the Zariski closure ofˉπ(G i)is connected and semisimple.If L′→L is an isogeny,thenˉπlifts to a mapˉπ′:G→L′since G i is simply connected.

Now assume that G i is the topological universal cover of G i(R).If k=R then any continuous homomorphism from G i to L is trivial,so we are done by the discussion above.If k=R thenπfactors through a continuous homomorphismˉπ:G i→L.The image ofˉπis a closed subgroup of L and so is the real points of a real algebraic subgroup. This implies thatˉπfactors through the covering map G i→G i(R).The conclusions of the lemma now follow as before.

3.2.α-invariant maps into algebraic varieties.Given two G-spaces S and Y,an L space R and a cocycleα:G×S→L,we call a map f:Y×S→Rα-invariant if f(gy,gs)=α(g,s)f(y,s)for all g and al-most every(y,s).Note that this de?nition di?ers slightly from the

LOCAL RIGIDITY FOR COCYCLES13 one in[Z2],where this map would be called?α-invariant where?αis the pullback ofαto G×Y×S.

The following theorem will play a key role in all proofs in this section. The assumption on the rank of G is only used to be able to apply this theorem.

Theorem3.5.Assume G acts weakly irreducibly on S preservingμ. Let M be the k points of an algebraic variety M de?ned over k on which L acts k rationally.Assume thatα:G×S→L is a Borel cocycle whose algebraic hull is L and that there exists a measurableα-invariant map φ:P\G×S→M such that the essential image ofφis not contained in the set of L-?xed points of M.Then there is a normal k-subgroup H

(1)p H?αis cohomologous to a continuous homomorphismπH:

G→L/H,where p H:L→L/H and H=H(k);

(2)L/H is semisimple and connected.

Proof.Letφs(x)=φ(x,s)where x∈P\G and s∈X.First one shows that eitherφs is rational for almost every s orφs is constant for almost every s.By rational we mean that there is i∈I such that k=k i,the mapφfactors through the projection p i:P\G→P i(k i)\G i(k i)which means thatφ=p i?ˉφwhere thatˉφis a k rational map P i\G i→M. Rationality ofφwas shown by Zimmer in[Z1]for irreducible actions with each G i=G i(k i)using an adaptation of an argument due to the second author[M1,M2].The proof goes through almost verbatim for weakly irreducible actions,as well as for the case where one G i is the universal cover of G i(R).See also pages104-5of[Z2]or[Fu3] for accessible presentations of special cases.Our de?nition of weak irreducibility is motivated by the ergodicity needed at this step of the proof.We now assume thatφs is rational and proceed in this case,the case ofφs constant is discussed at the end of the proof.

Secondly,one sees that the mapΦ:S→Rat(P\G,M)de?ned by Φ(s)=φs takes values in a single orbit.This follows from tameness of the G×L action on Rat(P\G,M)and the ergodicity of the G action on S,see the“Proof of Step3”on pages105-6and also Proposition 3.3.2of[Z2].

One now picks a rational mapψin this orbit and de?nes a map l:S→L such thatφs=l(s)ψ.Lettingβ(g,s)=l(gs)?1α(g,s)l(s) we have thatβ(g,s)ψ(x)=ψ(gx).Let H denote the point-wise stabi-lizer ofψ(P\G)in M.Since M=M(k)and L acts rationally on M, H=H(k)where H

14 D.FISHER AND G.A.MARGULIS

ψ(P\G)is L-invariant.Therefore H is normal in L,and H is normal in L.Fixing(almost any)s,and writingβs(g)=β(g,s),we have that βs(g1g2)ψ(x)=ψ(g1g2x)=βs(g1)ψ(g2x)=βs(g1)βs(g2)ψ(x).There-foreβs(g1g2)β(g2)?1β(g1)?1?xesψ(P\G)pointwise.It follows that p H?βs:G→L/H is a homomorphism.Thatπ=p H?βs is continuous follows from a result of Mackey,see[Z2,B.3].The remaining conclu-sions of the theorem follow from Lemma3.4.

Ifφis constant for almost every s∈S,we have anα-invariant map Φ:S→M.The image of this map is contained in a single H orbit since the L action on M is tame and the G action on S is ergodic. Since the L action on M is de?ned by an algebraic action of L on M, the stabilizer of this orbit is H=H(k)where H

3.3.Algebraic hull semisimple.We now prove Theorems1.4and Theorem1.5in the case where the algebraic hull of the cocycle is semisimple.

Theorem3.6.Let G act weakly irreducibly on S preservingμand let α:G×S→L be a Borel cocycle with algebraic hull L.Further assume that L is semisimple.Thenαis cohomologous to a cocycleβ=π·c. Hereπ:G→L is a continuous homomorphism and c:G×S→C is a cocycle taking values in a compact group C centralizingπ(G). Theorem 3.7.LetΓ

To reduce Theorem3.6to Theorem3.5we need to?nd a k variety M on which H acts k-rationally and anα-invariant map f:P\G×S→M.

LOCAL RIGIDITY FOR COCYCLES15 To produceαinvariant maps,one uses the following modi?cation of a lemma of Furstenberg from[Fu2]which can be deduced from Proposi-tions4.3.2,4.3.4and4.3.9of[Z2].The lemma holds under more general circumstances than those needed here.For the lemma,G can be a lo-cally compact,σ-compact group,P a closed amenable subgroup and L any topological group.

Lemma3.8.Assume G acts on S preservingμ.Letα:G×S→L be a Borel cocycle.Let B be any compact metrizable space on which L acts continuously and P(B)the space of Borel regular probability measures on B.Then there is anα-invariant map f:P\G×S→P(B).

We note here that we give a proof using only amenability of P, without reference to the notion of an amenable actions,though we do rely on ideas of Zimmer’s to construct a convex compact space on which P acts a?nely and continuously.

Proof.Let B be any compact L-space.Viaαwe can de?ne a skew product action of G on S×B.We consider the diagonal G action on G×S×B given by the right G action on G and the skew product action on S×B,which we note commutes with the left G action on G.LetμG be Haar measure on G and M(G×S×B)be the space of regular Borel measures on G×S×B which are invariant under the diagonal action and project toμG×μon G×S.We want to topologize M(G×S×B) so the left G action is continuous and the space is a compact con-vex a?ne 3ab8df0eba1aa8114431d975ing disintegration of measures,we can identify M(G×S×B)with F(G×S,P(B))the space of measurable maps from G×S to P(B).Let C(B)be the Banach space of continuous functions on B.We identify F(G×S,P(B))as a subset of L∞(G×S,C(B)?)and give L∞(G×S,C(B)?)the weak topology coming from the identi?ca-tion L∞(G×S,C(B)?)=L1(G×S,C(Y))?.In this topology the action of G is continuous and F(G×S,P(B))is a closed,convex subset of the unit ball in L∞(G×S,C(B)?).See[Z2,Section4.3]for more discussion of this and related constructions.It follows that there is a?xed point μP∈M(G×S×B)for the left P action.By applying disintegration of measures,this is left P-invariant,α-invariant map?f:G×S→P(B)or equivalently anα-invariant map f:P\G×S→P(B) We will also need the following lemma essentially due to Furstenberg. Lemma3.9.Let J0.

16 D.FISHER AND G.A.MARGULIS

For a proof,we refer the reader to[Z2,Lemma3.2.2]or the original article of Furstenberg[Fu1]in the case where k=R.

Proof of Theorem3.6.We call a representation of an algebraic group almost faithful if the kernel of the representation is?nite.We choose an almost faithful irreducible k-rational representationσof L on V such that the restriction ofσto any L0invariant subspace is almost faithful,where as usual L0denotes the connected component.(This can be done by inducing an almost faithful irreducible L0representation.) Let B=P(V)be the corresponding projective space.

Since P

Let J be the stabilizer of a pointμfor the L action on O.We prove that either J is compact or J is contained in an algebraic subgroup of positive codimension in L.If L is connected,this is Proposition 3.2.15of[Z2].By Lemma3.9,if J is not projectively compact,then there is a proper subspace W0.Since L is semisimple and the representation on V is almost faithful,the map from L to P GL(V)has?nite kernel and only compact subgroups of L are projectively compact.Assuming J is non-compact,we choose W of minimal dimension among subspaces with positiveμmeasure. Since the measure of W is positive,J·W must be a?nite union of disjoint subspaces∪n l=1W l,and we let F be the stabilizer of the J orbit J·W.Let F=F(k).The stabilizer J W in J of W is of?nite index in J and,by minimality of W and Lemma3.9,acts on P(W)via a homomorphism to a compact subgroup of P GL(W).If dim(F)= dim(L),then the connected component of L preserves∪n l=1W l and by connectedness preserves W.Since J W

If J is compact,then Lemma5.2.10of[Z2]applies and shows that the cocycleαis cohomologous to one with bounded image.If J

LOCAL RIGIDITY FOR COCYCLES17 This theorem produces a normal k-subgroup of positive codimension H

3.4.Conditional results using G-integrability ofα.In this sec-tion we prove a conditional result concerning the algebraic hull of G-integrable cocycles.The assumption of G-integrability is only used here.

Before stating our result,we?x some notation and assumptions.We assume that G has property T and that G acts weakly irreducible on (S,μ).Letα:G×S→L be a G-integrable Borel cocycle and assume that L is the algebraic hull of the cocycle.We can write L=F?U where F and U are k-subgroups,U is the unipotent radical of L and F is reductive.Let p F:L→F and be the natural projection.We assume that the cocycle p F?αis cohomologous to a cocycle of the formπ·c whereπ:G→F is a continuous homomorphism and c is a cocycle taking values in a compact subgroup C

Theorem3.10.Under the hypotheses discussed in the preceding para-graph,U commutes withπ(G).

18 D.FISHER AND G.A.MARGULIS

We prove the theorem by contradiction.The general scheme is as follows.If U does not commute withπ(G)there exists a k-rational action of L on a variety M and anα-invariant mapφinto M(k)such that the pointwise stabilizer H=H(k)of the image does not contain all of U.Applying Theorem3.5we obtain a contradiction,since number 2of that theorem implies that L/H is semisimple and this implies that U

We will construct a measurable mapφthat satis?es the hypotheses of Theorem3.5by using Oseledec’multiplicative ergodic theorem.We will give an argument that is close to the one in[M3,Section V.3-4], but also refer the reader to[Z4]for a somewhat di?erent approach. Let l be the Lie algebra of L.Let Gr j(l)be the Grassmann variety of j planes in l.We have an action of L on l by the adjoint representation which also de?nes an action of L on Gr j(l).

We look at the representation Ad l?π.

Theorem3.11.Assumeπ(G)does not commute with U.Then there is an integer0

Before proving Theorem3.11,we show how that result implies Theorem 3.10.

Proof of Theorem3.10.We now apply Theorem3.5to the mapφfrom Theorem3.11.This is possible since the stabilizer of the essential image does not contain U and so the essential image is not contained in L?xed points.If H is the stabilizer of the essential image ofφthis implies that L/H is semisimple and therefore that U

Lemma 3.12.Let A be an automorphism of(S,μ)and f a non-negative measurable function on X.Then for almost all x∈X

1

lim inf

m→∞

f(A m(x))=0.

m

LOCAL RIGIDITY FOR COCYCLES19 Let A be an ergodic automorphism of(S,μ)and W be a k vector space.Let u:Z×S→GL(W)be a Z-integrable cocycle over the Z-action generated by A.If we de?ne

1

χ+(u,x,w)=lim

m→∞

ln u(A m,x)w)

m

it follows from Oseledec multiplicative ergodic theorem thatχ+(u,x,w) andχ?(u,xw)exist for almost all x∈X and all w∈W.Furthermore, that theorem shows that there exists a?nite set J and real numbers χj(u)and mapsωj(u,x):S→Gr l(j)(W)such that

(1)for almost all x∈S,the space W is the direct sum⊕Jωj(u,x);

(2)for almost all x∈S the sequence{1

20 D.FISHER AND G.A.MARGULIS

a characteristic subspaceωl(j)(u,x)such thatωl(u V,x)=p(ωl(j)(u,x)) (respectivelyωl(u Q,x)=ωl(j)(u,x)∩Q).

For certain cocycles it is easy to compute characteristic subspaces and numbers.Letσ:Z→GL(W)be a homomorphism,let M=σ(1)and let c:Z×S→GL(W)be a cocycle taking values in a com-pact subgroup of GL(W).We let u(m,x)=σ(m)c(m,x).We let ?(M)be the set of all eigenvalues of M,Wλ(M)the eigenspace cor-responding toλ∈?(M)and W d(M)=[⊕ln|λ|=d Wλ(M)]k.We also let W+(M)=⊕d>0W d(M)and W≤0(M)=⊕d≤0W d(M).We will call d a characteristic number of M and W d(M)a characteristic subspace of M.Then the characteristic numbers of u are the characteristic num-bers of M and the characteristic subspaces for u are the characteristic

numbers for M.Furthermore the space⊕χ

j≤0ωj(u,x)=W≤0(M)and

⊕χ

j>0

ωj(u,x)=W+(M)

Proof of Theorem3.11.As the proof is very involved,we divide it into several steps.The basic idea is to choose an element t of G and use Oseledec theorem to construct characteristic maps from S→Gr m(l)for αand each g?1tg.This gives anα-invariant mapφ:G×S→Gr m(l), which we show descends to anα-invariant mapφ:P\G×S→Gr m(l). We then pass to characteristic subspaces for the cocycleα′which is co-homologous toαand where p F?α′=π·c′.We use the functoriality of characteristic subspaces and the form ofα′to compute the characteris-tic subspaces quite explicitly.Finally using the assumption thatπ(G) does not commute with U,we show that the stabilizer of the essential image does not contain U.

Step One:Choosing t.

We call a subgroup diagonalizable if it can be conjugated to a sub-group of the group of diagonal matrices.Recall that a subgroup S i

(1)the group generated by t is not contained in any proper normal

subgroup of G

(2)for anyχ∈X(T i)whereχ(τi(t))has modulus one it follows that

χ(τi(t))=1.

Proof.To satisfy1,it su?ces to choose t such that it projects to an element which generates an in?nite subgroup in each simple factor of each G i.

LOCAL RIGIDITY FOR COCYCLES21 For k i=R it su?ces to assume thatχ(τi(t))is positive for every χ∈X(T i).If k i is non-Archimedean,we identify T i with(k?i)l(i)where l(i)is the k i-rank of G i.We chooseπa uniformizer of k i.We assume

that the projection ofτi(t)to each copy of k?i in(k?α

0)i(α0)is the product

of a unit of k?i with a non-zero power ofπ. Remark:Letπbe?nite dimensional representation of G on a vector space V.It follows from our choice of t that ifπ(t)has all eigenvalues of modulus one thenπis trivial.

Step two:Oseledec theorem and characteristic maps.

We will construct the mapφ:G×S→Gr k(h)by applying Oseledec theorem to certain cocycles over the action of g?1tg on S.

Since G acts ergodically on S,it follows from the Mautner phe-nomenon that t and therefore g?1tg does as well[M3,II.3.3].We de?ne a mapφ′:G×S→Gr k(l).The element g?1tg generates a Z ac-tion on S.We de?ne a cocycle u g:Z×S→GL(l)over this Z action by u g(m,x)=Ad lα(g?1t m g,x)and apply Oseledec theorem to each cocycle u g.Since di?erent choices of g∈G de?ne cohomologous cocy-cles over conjugate actions,it follows that the characteristic numbers χj(u g)do not depend on g nor do the dimensions l(j)of the subspaces ωj(u g,x).Weχj=χj(u g)andωj(g,x)=ω(u g,x).We now have maps ωj:G×S→Gr l(j)(l).If we let G on G×X by h(g,x)=(gh?1,hx),the mapωj:G×X→H isα-invariant.To show this one uses the cocycle identity to see that

α(hg?1s m gh?1,hx)=α(h,g?1s m gx)α(g?1s m g,x)α(h,x)?1

and notes that

lim inf

m→∞

1

本文来源：https://www.bwwdw.com/article/fbne.html