Continuous Quotients for Lattice Actions on Compact Spaces

Continuous Quotients for Lattice Actionson Compact Spaces

DAVID FISHER1 and KEVIN WHYTE2

1Department of Mathematics, Yale University, P.O. Box 208283, New Haven,CT 06520-8283, U.S.A. e-mail: david.¢[email protected] of Mathematics, University of Chicago, 5734 S. University Ave, Chicago, Il60637, U.S.A. e-mail: [email protected]

(Received: 22 March 2000)

Abstract. Let G < SLn�Z� be a subgroup of ¢nite index, where nX 5. Suppose G acts continu-ously on a manifold M, where p1�M� � Zn, preserving a measure that is positive on open sets.Further assume that the induced G action on H1�M� is non-trivial.We show there exists a ¢niteindex subgroup G0 < G and a G0 equivariant continuous map c : M ! Tn that induces anisomorphism on fundamental group. We prove more general results providing continuousquotients in cases where p1�M� surjects onto a ¢nitely generated torsion free nilpotent group.We also give some new examples of manifolds with G actions.

Mathematics Subject Classi¢cations (2000). 37C85, 53C24.

Key words. large group actions, rigidity, ergodic theory and semisimple groups.

1. Introduction

Let G be a semisimple Lie group with R-rank�G�X 2, and G < G a lattice. In thispaper we seek to examine the relationship between the topology and dynamicsof measure preserving actions of G on a compact manifold M. More precisely,we seek relations between the fundamental group of M and the structure of bothM and the G action on M. The simplest version of our main theorem is:

COROLLARY 1.1. Let G < SLn�Z�; nX 3 be a subgroup of ¢nite index. Suppose Gacts on a compact manifold M preserving a measure that is positive on open sets.Assume r : p1�M� ! Zn is G equivariant, that the action of G on Zn is given bythe standard representation of SLn�R�, and that theG action lifts to ker r. (The liftingcondition is automatic provided nX 5). Then there is a ¢nite index subgroup G0 < Gand a G0 equivariant map c : M! Tn which induces the map r on fundamentalgroups.

The actual result applies more generally to certain kinds of actions (describedprecisely below) of lattices on manifolds whose fundamental group surjects ontoa torsion free ¢nitely generated nilpotent group. In all cases we produce a continuous

Geometriae Dedicata 87: 181^189, 2001. 181# 2001 Kluwer Academic Publishers. Printed in the Netherlands.

map from our action to an algebraically de¢ned action on a nilmanifold. In par-ticular, the theorem applies to the surgery examples of Katok and Lewis, andthe quotient recaptures the torus action on which the surgeries were performed [KL].For some of the examples with more complicated fundamental group it will benecessary to pass to a ¢nite cover before our theorem applies. For all of theseexamples, the map to the torus simply collapses certain invariant submanifoldswhose fundamental groups map to 0 in Zn. In Section 4, we prove a purelytopological result that shows that this is essentially what all such maps must looklike.

This work is related to work of the ¢rst author, who shows in [F] that under dif-ferent hypothesis there are measurable maps to the same standard examples.The argument of [F] also shows that fundamental groups of manifolds with measurepreserving G actions are of arithmetic type under fairly mild hypotheses relating thedynamics to the topology, unless the fundamental group admits no linearrepresentation. Further work of the ¢rst author with Zimmer [FZ] produces linearrepresentations of the fundamental group under stronger geometric assumptions.Combining these results gives that either the action on fundamental group is trivialor that the fundamental group contains a direct factor that is ¢nitely generatedand nilpotent on which G acts nontrivially. These results give some evidence thatour assumptions are not atypical, at least in the case that G! �Out�p1�M�� isnontrivial.

We also discuss several examples that show that for our general result one cannotexpect to improve the regularity of the quotients without stronger assumptions. Notethat our theorems apply not just to manifolds but to any topological spaces satisfyingstandard covering space theory, i.e. any connected, locally path-connected,semi-locally 1-connected, locally compact, separable, metrizable space with ¢nitelygenerated fundamental group.

2. Preliminaries

We assemble here some basic facts from topology and dynamics that will be used inthe proof.

Suppose a ¢nitely generated group G acts continuously on a manifoldM. Let ~M beany cover ofM and D and D be the deck group of ~M overM and the group of lifts ofthe G action, respectively. There is an exact sequence:

1! D! D! G! 1

and the G action lifting to ~M is equivalent to this exact sequence splitting. In par-ticular, elementary group cohomology shows that the sequence splits ifH2�G;D� � 0 and the map G! Out�D� given by the sequence lifts to a mapG! Aut�D�.

Suppose G, a group, acts on a measure space M preserving a ¢nite measure. Acocycle is a measurable map a: G�M! H where H is a group and a satis¢es

182 DAVID FISHER AND KEVIN WHYTE

the equation a�g1g2;m� � a�g1; g2m�a�g2;m� for all g1; g2 2 G and all m 2M. Twococycles a and b are cohomologous if there is a measurable map f: M ! H suchthat a�g;m� � f�gm�ÿ1b�g;m�f�m� for all g 2 G and almost all m 2M. Now assumethat G < G is a lattice where G is a semisimple Lie group, that the G action onM is ergodic, and thatH is an algebraic group. By results of Zimmer, for any cocyclea: G�M ! H, there is a minimal subgroup L < H, unique up to conjugacy, suchthat a is cohomologous to a cocycle taking values in L. The group L is calledthe algebraic hull of the cocycle [Z1].

We will also need the following de¢nition:

DEFINITION 2.1. A representation s: G! GLn�R� is weakly hyperbolic if there isno invariant subspace V < Rn where all the eigenvalues of all elements of G havemodulus one.

3. Main Theorem and Proof

Suppose G, a ¢nitely generated group, acts on a compact manifold M and there is asurjection r: p1�M� ! L, whereL is a ¢nitely generated torsion free nilpotent group.

If the action of G lifts to the cover of M corresponding to ker r, then we have amap s: G! Aut�L�. By a theorem ofMalcev, this gives a map s: G! Aut�N�whereN is a simply connected nilpotent Lie group, containing G as a lattice. We can view sas a representation s: G! GL�n� where n � Lie�N�.

DEFINITION 3.1. Under the conditions described above, we say that the G actionon M is p1-hyperbolic if s: G! GL�n� is weakly hyperbolic.

Note that our de¢nition of p1 hyperbolic includes the assumption that the G actionlifts to the appropriate cover of M.

THEOREM 3.2. Let G < G be an irreducible lattice, where G is a semisimple Liegroup with all simple factors of real rank at least two. Suppose that G acts on acompact manifold M preserving a measure that is positive on open sets. Assume thereis a surjection r: p1�M� ! L where L is a ¢nitely generated torsion free nilpotentgroup and that the G action is p1-hyperbolic. Then there is a ¢nite index subgroupG0 < G and a G0 equivariant map c: M ! N=L where the action on N=L is de¢nedby extending the action of G0 on L to N. Furthermore, the map c� on fundamentalgroup is equal to r above.

In the case ofM having abelian fundamental group, this gives Corollary 1.1 from theIntroduction. We need only explain the remark that the lifting of the action is auto-matic in the case where nX 5. To see this, we need only see that H2�G;Zn� � 0,since Aut�Zn� � Out�Zn�. By Theorem 4.4 of [B], we know that H2�G;Rn� � 0.By looking at the long exact sequence in cohomology corresponding to

CONTINUOUS QUOTIENTS FOR LATTICE ACTIONS 183

1! Zn ! Rn! Tn ! 1, this implies that H2�G;Zn� � H1�G;Tn� which is ¢niteand vanishes if we pass to a subgroup of ¢nite index. This allows us to lift the actionof this subgroup of ¢nite index, which is suf¢cient for our purposes. Note that thestatement in the abstract follows from the corollary, since Margulis' superrigiditytheorem and the action on H1�M� being nontrivial imply that the action on Zn

is indeed given by the standard representation.Proof. Since r: p1�M� ! L is a surjection, we have a continuous map

f : M ! N=L. This follows since N is contractible and N=L is anEilenberg^MacLane space for L and hence M has a continuous map to N=Linducing r on fundamental groups which is canonical up to homotopy.

We can lift f to a L equivariant map ~f : ~M ! N where ~M is the cover of Mcorresponding to ker r.

We consider the map ~a: G� ~M ! N de¢ned by

~a�g;m� � ~f �gm��gf �m��ÿ1:This is clearly a measure of the extent to which ~f fails to be equivariant, and is de¢nedsince we have assumed the G action lifts to ~M. First we show that ~a descends to a mapa: G�M ! N. To see this, let m 2 ~M and ml be any translate of m where l 2 L isviewed as a deck transformation of ~M over M. It suf¢ces to show thata�g;m� � a�g;ml�, but

a�g;ml� � ~f �gml��gf �ml��ÿ1

� ~f ��gm��gl��g�f �m�l��ÿ1

� ~f �gm��gl��gf �m��gl��ÿ1

� ~f �gm��gl��gl�ÿ1�gf �m��ÿ1

� ~f �gm��gf �m��ÿ1

since ~f isL equivariant and the action of G onL induced by the action on p1�M� is thesame as the action of G on L < N.

We now look at the map b: G�M ! G N de¢ned by b�g;m� � �g; a�g;m��. Asimple computation veri¢es that b is a cocycle over the G action on M. We can viewb as a cocycle into G N by the natural inclusion. For now, we assume the actionis ergodic. By results of Lewis and Zimmer, the algebraic hull, L, of this cocyclewill be reductive with compact center. Let L0 < L be the connected componentof the identity in L. By passing to a ¢nite ergodic extension of the action onX �M � L=L0 we have a cocycle b: G� X ! G N (still called b) with algebraichull L0. Note that b�m; l� depends only on m. Since any connected reductivesubgroup of G N is conjugate to a subgroup of G, we can assume that L0 < G.This means that the cocycle on all of X is cohomologous to one taking valuesin G, in other words b�g; x� � f�gx�ÿ1d�g; x�f�x� where f: X ! G N is a


measurable map and d: G� X ! G N is a cocycle taking values entirely inG. Writef�x� � �f1�x�;f2�x�� f1�x�; 1N��1G;f2�x��. Since d � �d1; 1N�, computing thecocycle equivalence above in components yields that

b�g; x� � �1G;f2�gx�ÿ1�g; 1N ��1G;f2�x��:We now show thatf2 is continuous. The argument follows exactly as in Lemma 6.5

of [MQ]. For the reader's convenience we repeat here the case where N � Rn.Essentially the idea is to use the fact that Rn is spanned by contracting directionsfor elements of G and to show that along any contracting direction, f2 can asthe limit of iterated contractions of a.

For g 2 G let E�g� and F �g� be subspaces ofRn that are the generalized eigenspacesof g with eigenvalues of absolute value > 1 and W 1 respectively. ClearlyRn � E�g� � F �g�, and the assumption of weak hyperbolicity implies that Rn isspanned by fE�g� j g 2 Gg. We show continuity of f2 by showing continuity off2 projected onto any E�g�. For any function h: M! Rn we write hE�g� for the com-position of the function with projection on E�g�.

Looking at what the cocycle condition on b implies for a we see thatf2�x� � gÿ1f2�gx� � ga�g;m� where x � �m; f � 2 X . Iterating this equality andprojecting to E�g� gives

f2E�g��x� �Xni�1�gi�ÿ1jE�g�aE�g��g; giÿ1m� � gnÿ1jE�g�f2E�g��gnx�:

Since the eigenvalues of gÿ1jE�g� all have absolute value < 1, on a set of full measuregnÿ1jE�g�f2E�g��gnx� ! 0 as n!1. So we have:

f2E�g��x� �X1i�1�gi�ÿ1jE�g�aE�g��g; giÿ1m� ��

which converges uniformly since a�g;ÿ� is continuous and bounded function on M.This shows both that f2 is continuous and is a function on M that is independentof the ¢nite ergodic extension X .

If the action is not ergodic, we simply carry out the analysis above on each ergodiccomponent. We will get a function f2 as above for each component. Since �� aboveshows how to compute f2 explicitly on any ergodic component, we see that f2

is a well de¢ned continuous function from X to Rn. From �� it is clear that f2

descends to a continuous function M to Rn. When Rn is replace by a more generalsimply connected nilpotent group, the analysis becomes more complicated butfollows exactly as in [MQ].

Now b�g;m� � �g; ~f �gm��g ~f �m��ÿ1� where we are actually choosing some lift of thepoint m to ~M. Substituting this in above and computing the N factor gives~f �gm��g ~f �m��ÿ1 � f2�gm�ÿ1�f2�m��. We can lift f2 to a map from ~M to N, and thenrearranging the last expression gives ~f2�gm�~f �gm� � g ~f2�m�g~f �m� � g� ~f2

~f �. Thisshows that the map � ~f2��~f �: ~M ! N is G equivariant. Since ~f2 is L invariant, ~f


is L equivariant and L acts on the right on ~M, we see that the map f2f : M! N=L isalso G equivariant. Note that f2: M! N is a map into a contractible space and soisotopically trivial. This implies that f2f is in the same isotopy class as f andtherefore that f2f induces the map r on fundamental group. &

4. Examples

For the sake of clarity we discuss the case of Corollary 1.1 only, although much caneasily be generalized to the general case of Theorem 3.2. Throughout G will referto a ¢nite index subgroup of SLnZ.

We start with some non-trivial examples to which our theorem applies. Theseexamples of exotic SLn�Z� actions are due to Katok and Lewis ([KL]). They are con-structed from the standard action on Tn by blowing up the ¢xed point, so that itbecomes a copy ofRPnÿ1 with the standard action of SLnZ.What is not at all obviousthat the resulting manifold has an invariant analytic structure and volume form.Indeed, in order tomake the action preserve a volume form onemust use a differentialstructure which is not the obvious one. The continuous map to the torus, collapsingthe RPnÿ1, is not smooth with respect to this new smooth structure. It is immediatefrom the proof of Theorem 3.2 that the map is unique, since it is given explicitlyby (*) in terms of the action. This shows that the regularity of the semi-conjugacyin Theorem 3.2 cannot be improved, even when the action is analytic.

There are further examples of exotic actions of SLn�Z�, which were originallyconstructed by Weinberger. Take Tn and remove some ¢nite invariant set. Theresulting manifold can be compacti¢ed to a manifold with boundary by addingthe spheres in the tangent bundle at each point. The action of G on the boundaryis the standard linear actions. These actions extend over n-balls - just think ofthe ball as Rn with the linear G action as the action on the sphere at in¢nity. Gluingin these balls gives a closed manifold with a G action. The underlying manifoldis still the torus, but the action is different. There is no invariant measure onthe whole torus, but there is on the complement of the disks. Thus our theoremsays there is a continuous map from this complement to the torus. Indeed themap which simply collapses the balls we glued in back to points is continuousand equivariant.

It is possible to combine Weinberger's construction with some surgery to producea new class of examples. Take �X ; @X � any compact manifold with boundary, letN beWeinberger's example cross a @X . Inside of N is a ball cross @X . Remove the interiorof this, and glue in X cross an nÿ 1 sphere in its place to get M. If p1�X � is trivial,p1�M� � Zn. As a speci¢c example, taking X to be the m ball gives an exampleof a G-manifold with p1 � Zn but which is de¢nitely not, even non-equivariantly,a bundle over Tn.

In the above examples the map to the torus is always well behaved off of asubmanifold which is collapsed. The following lemma, in some ways a converseto the main theorem, shows that this collapsing must occur in general.


LEMMA 4.1. Let Z be any compact, connected, space with a G action. Anynon-constant equivariant map to the torus surjects p1�Z� onto a ¢nite index subgroupof Zn.

Proof. Suppose not. Then, since the image of p1�Z� is a G invariant in¢nite indexsubgroup of Zn, it must be trivial. Thus we aim to show that any null homotopicmap is constant. Let f : Z! Tn be a null homotopic equivariant map.

The image of f in Tn is closed, invariant, and connected. Hence, since we assumethe map non-constant, it must be surjective. In particular, some z0 in M mapsto 0 in Tn. Lift f to a map F : Z! Rn such that F �z0� � 0.

Fix a g 2 G. Since F covers an equivariant map, we know that tg�z� � F �gz� ÿ gF �z�takes values inZn. Since it is continuous in z, this implies it is constant in z. We de¢netg to be the common value. It is easy to see that t: G! Zn is a 1-cocycle, in otherwords tgs � tg � gts.

We can evaluate tg at z0, which yields tg � F �gz0�. Since the image of F is compact,this shows that tg is bounded independant of g. The cocycle identity then showsthat gts is bounded independantly of g and s. In particular, ts has bounded Gorbit and so we have ts � 0 for all s. This means precisely that F is equivariant.This ¢nishes the proof as F �Z� is then a bounded, invariant set in Rn, and thusis f0g. &

In all of these examples, the map to the torus is nice everywhere except thepre-image of a ¢nite invariant set. We believe that will always be the case. Notethat if the map had any regularity, then by Sard's theorem the critical values wouldbe measure zero. Since they are closed and invariant this would limit them to a ¢niteinvariant set, and thus we would know that off this ¢nite set our manifold is a bundleover the torus. We know from the Katok^Lewis examples that the map need not beC1, even for analytic actions. The map in that case is, however, still analytic offa lower dimensional submanifold.

QUESTION 4.2. If, in the statement of the main theorem, one assumes the action tohave some regularity, is the map also regular away from the pre-image of a ¢niteinvariant set in the torus?

Even if regularity does not hold, one can still hope the map must be ``taut'' in somesense. The last lemma is one example of the sort of substitute for regularity. At thevery least one would like to be able to rule out space ¢lling curves in this context,so as to prove:

CONJECTURE 4.3. Any compact manifold with p1 � Zn and a G action whichinduces the standard action on p1 must be of dimension at least n.


Here we have no assumption of an invariant measure. In all the examples there isan invariant measure, at least on a large open set. Is this always the case?

QUESTION 4.4. If SLn�Z� acts on a compact manifold with fundamental group Zn,inducing the standard action on p1, is there always an invariant measure? Is therealways an invariant measure whose support contains an open set?

One is tempted to view the torus with the standard action as some kind ofequivariant classifying space for G actions with p1 � Zn, and to view our theoremas proving the existence of a classifying map One cannot hope for general resultsof this type ^ the use of superrigidity is not just an artifact of the method. Evenwhen there is a clear candidate classifying space, and the action there is hyperbolic,the analog of our theorem need not hold.

Consider the action of SL2�Z� on T2. Since SL2 acts hyperbolically on the torus,one might expect our theorem to cover this case. It does not: take G any torsionfree subgroup of ¢nite index in SL2�Z�. Such a G is free. Starting with the standardaction of G on T2, conjugate the action of one of the free generators by ahomeomorphism homotopic to the identity, and leave the action of the remaininggenerators unchanged. Since G is free, this still generates an action of G. Thereis no equivariant map of this torus to the standard one. This follows from theuniqueness of the map conjugating a single Anosov homeomorphism to a linearAnosov automorphism, which follows from the same reasoning that shows themap in Theorem 1.1 is unique (or see [KH]).

References

[B] Borel, A.: Stable Real Cohomology of Arithmetic Groups II, Progr. Math. 14 (1981).[F] Fisher, D.: On the arithmetic structure of lattice actions on compact manifolds,

preprint.[FZ] Fisher, D. and Zimmer, R.: Geometric lattice actions and fundamental groups,

preprint.[KH] Katok, A. and Hasselblatt, B.: Introduction to the Modern Theory of Dynamical

Systems, Cambridge Univ. Press, Cambridge 1995.[KL] Katok, A. and Lewis, J.: Global rigidity for lattice actions on tori and new examples of

volume preserving actions, Israel J. Math 93 (1996), 253^280.[LZ1] Lubotzky, A. and Zimmer, R. J.: Arithmetic structure of fundamental groups and

actions of semisimple groups, Preprint.[MQ] Margulis, G. and Qian, N.: Local rigidity of weakly hyperbolic actions of higher real

rank semisimple Lie groups and their lattices, to appear in Ergodic Theory Dynam.Systems.

[W] Witte, D.: Measurable quotients of unipotent translations on homogeneous spaces,Trans. Amer. Math. Soc. 354(2) (1994), 577^594.

[Z1] Zimmer, R. J.: Ergodic Theory and Semisimple Groups, BirkhÌuser, Boston, 1984.[Z2] Zimmer, R. J.: Actions of semisimple groups and discrete subgroups, Proc. Internat.

Congr. Math., Berkeley, 1986, 1247^1258.


[Z3] Zimmer, R. J.: Lattices in semisimple groups and invariant geometric structures oncompact manifolds, In: Roger Howe (ed.), Discrete Groups in Geometry and Analysis,BirkhÌuser, Boston, 1987, 152^210.

[Z4] Zimmer, R. J.: Representations of fundamental groups of manifolds with a semisimpletransformation group, J. Amer. Math. Soc. 2 (1989), 201^213.


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Continuous Quotients for Lattice Actions on Compact Spaces