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Discrete Groups, Symmetric Spaces, and Global Holonomy Author(S): Joseph A

Discrete Groups, Symmetric Spaces, and Global Holonomy Author(S): Joseph A

Discrete Groups, Symmetric Spaces, and Global Author(s): Joseph A. Wolf Reviewed work(s): Source: American Journal of , Vol. 84, No. 4 (Oct., 1962), pp. 527-542 Published by: The Johns Hopkins University Press Stable URL: http://www.jstor.org/stable/2372860 . Accessed: 03/05/2012 14:57

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http://www.jstor.org DISCRETE GROUPS, SYMMETRIC SPACES, AND Gl,OBAL HOLONOMY.*

By JOSEPH A. WOLF.'

1. Introduction. Let M be a connectedsimply connected Riemannian manifoldand let r be a properlydiscontinuous of isometriessuch that M/r is compact. If everysectional curvature of M is negative,in particularif M is a noncompactirreducible of rank 1, thena methodof iR. Cartanshows that every abelian subgroupof r is eitherfinite or the productof a finitegroup with an infinitecyclic group. If M is the Rn, then a calculationshows that everyabelian subgroupof r is the productof a finitegroup with a freeabelian groupon ? n generators. These phenomena are unifiedby one of the conclusionsof our Theorem6. 2: If M is Riemannian symmetricand v is the maximumof the dimensionsof thosetotally geodesic submanifoldsof M whichare isometricto Euclidean spaces,then every abelian subgroupof r is the productof a finitegroup with a free on < v generators,and r has a subgroupwhich is free abelian on v generators. We also provethat an abelian subgroupof r must preservea flat connected totallygeodesic submanifold of M; if M/r is a manifold,it followsthat M/r containsa maximalconnected flat totally geodesic submanifold which is closed, and everyabelian subgroupof -r,(Mr/) can be representedby closed geodesic arcs lying in a connectedflat totallygeodesic submanifold(Corollary 6. 6). In addition,we analyzethe groupof componentsof the homogeneousholonomy group of a locallysymmetric N (Theorem 7. 1), prove thatN has compacthomogeneous holonomy group if N is compact2 (Corollary 7.2), and give conditionsfor everymanifold locally isometricto N to have compacthomogeneous holonomy group (Corollary7. 3). Our bounds are obtainedby estimatingthe "size" of abelian subgroups of discreteuniform subgroups of Lie groupsL =E X G whereE is a semi- directproduct of a compactgroup and a vectorgroup, such as the ,and G is a reductiveLie group with only finitelymany components.

* Received October 26, 1961; revised August 4, 1962. 1 The author thanks the National Science Foundation for fellowship support during the preparation of this paper. 2 If N is flat, this is just the classical Bieberbach Theorem [3]. This does not give a new proof of the Bieberbach Theorem because that result is used in our arguments.

527 528 JOSEPH A. WOLF.

The estimatesare firstmade for reductivegroups (Theorem 4. 2), and then extendedby a generalizationof Bieberbach'sTheorem (Theorem 5. 1). Let r be a discreteuniform subgroup of a reductiveLie group G, where G has onlyfinitely many components, and let A be an abelian subgroupof r. Our main idea is that the size of A can be estimatedby findinga Cartan subgroupH of G which is normalizedby A, and observingthat A n H has finiteindex in A. In orderto findH, we firstprove that everyelement of r is a semisimpleelement of G (Theorem 3. 2), and then apply a result of A. Borel and G. D. Mostow (Corollary3. 7). The main tool in our proof of Theorem3. 2 is a geometriccharacterization of the semisimpleelements of G (Lemma 3. 6).

2. Preliminaries. 2.1 Lie groups. Given a G, Go will denote the identity component,(M will denotethe , and exp: S -> G will denotethe exponentialmap. G and (Mare called reductiveif the of (M (or, equivalently,of Go) is fullyreducible, i. e., if (Mis the directsum of an abelian ideal SCand a semisimpleideal WY;then SCis the centerof (M, A = exp(C) is the identitycomponent of the centerof Go and is called the connectedcenter of G,,,(' is the derivedalgebra of S and is called the semi- simplepart of (M,G' exp(E') is called the semisimplepart of G,, and there is a natural homomorphism(a, g) -> ag of A X G' onto Go. If S is reductive,then the Cartan subalgebrasof (M are the subalgebras of the formSC 0 &', where0 denotesdirect sum of ideals and &' is a Cartan subalgebraof WY;thus the Cartan subalgebrasof (Mare abelian. By Cartan subgroupof a Lie group G, we mean a (necessarilyconnected) group of the formexp (.e) where& is a Cartan subalgebraof (S. Under the adjoint representationof a Lie group G, an elementg C G inducesan automorphismad(g) of (M; we will call g semisimpleif ad(g) is a fully reduciblelinear transformationof (M. If g is a semisimpleelement of a reductiveLie group G, and if Z is the centralizerof g in G, then not onlyis Z reductivebut the adjoint representationof G inducesa fullyreducible representationof 8 on (S. If S and T are subsets of a group G, then the commutator[s, T] denotesthe set of all elements[s, t] sts-'t-1where s C S, t C T. If S and T are groupsand q is a homomorphismof S into the groupof automorphismsof T, then the semidirectproduct S pT is the set S X T with the group structure (s1,t,) (s2,t2) = (s1s2, (cp(s2-) (t,))t2). If S is DISCRETE GROUPS. 529 given as a group of automorphismsof T, then the semidirectproduct is denotedS. T. 2. 2. Discretegroups. A subgroupr of a topologicalgroup G is discrete if G has an U such that r n u is just the identityelement 1 E G. A subgroupH of G is uniformif the coset space G/I (I7 is the closureof H in G) is compact. Let r be a discreteuniform subgroup of G. C. L. Siegel [9] has shown that G is locally compact,and, if everycovering of G by open sets has a countablerefinement,3 then G has a compact subsetF such that r- F= G, every g E G has a neighborhoodcontained in a finite union of the yF, rF = {y C r: yF meets F} is finite, and rF generates r if G is connected. It followsthat P finitelygenerated if G/Gois finitelygenerated, but this is betterseen directly[7]. F will be called a fundamentaldomain for the action of r on G by righttranslations.

2. 3. Symmetricspaces. It is well known that a connectedsimply connected Riemannian symmetricspace M is isometric to a product MOXX lXX . X Mt whereMO is a Euclidean space and each M, (i > 0) is an irreducibleRiemannian (non-Euclideanand not isometricto a product of lower dimensionalRiemannian manifolds) symmetricspace; Mo is the Euclidean part of M, ' =-1 X . . . X Mt is the non-Euclideanpart of M3, and the MT (i > 0) are the irreduciblefactors of 3. We will say that M is strictlynon-Euclidean if M M=M', i. e., if dim.MO = 0, and will say that M is strictlynoncompact if everyirreducible factor of M is noncompact. If M is strictlynoncompact, then everysectional curvature on M is ? 0. Full groups of isometriesare related by I(M) =I((MO) X I(M), and I (M') is generatedby I (M1) X . . . X I (Mt) togetherwith all permutations on mutuallyisometric sets of Mi. Connectedgroups of isometriesare related by Io (M) = Io (Mo) X Io (Ml) X . . . X Io (Mt), and I (M)/Io(M) is finite. I (Mo3)is the Euclideangroup E (dim.Mo ), I (MI) is a compactsemisimple Lie groupif M,,is compact,and Io (Mi) is a noncompactcenterless real or complex simpleLie groupif M, is noncompact(i > 0). Let S be a maximalconnected flat (all sectionalcurvatures zero) totally geodesic submanifoldof M. I(M) acts transitivelyon the set of all such submanifolds,and the rank of M (denoted rank.M) is definedto be their commondimension. S is isometricto a productMO X S, X . *X St where Si is a maximalconnected flat totally geodesic submanifold of Mi, whenceit

3 Siegel requires that G have a countable basis for open sets, but uses only this weaker property. 530 JOSEPH A. WOLF. is easily seen that S is a closed submanifoldof M. The symmetryto MI at a point of Si induces a symmetryof Si; it followsthat each Si (and thus S) is a connectedRiemannian symmetric manifold of constantcurvature zero. Thus Si is a flat torus if Mi is compactand Si is a Euclidean space if Mi (i> 0) is noncompact;to see this, we use [10, Theoreme4] or [11, ? 14] togetherwith the factthat Si is the orbitof some Cartansubgroup of maximal vectorrank in I (Mj). If MC is the productof the compactirreducible factors of M and MN is the productof MO withthe noncompactirreducible factors of M, it followsthat S is isometricto the productof a flattorus of dimension rank.Mc with a Euclidean space of dimensionrank.MN. For this reason, we definethe vectorrank of M (denotedv-rank. M) to be rank.MN. Observe that v-rank.M - dim.MO + v-rank.M'. Let r be a subgroupof I (M). The action of r on M is properlydis- continuousif everyelement of M has a neighborhoodwhich meets its trans- formsby only a finitenumber of elementsof r; this is equivalentto r being a discretesubgroup of I (M). The action of r on M is freeif 1 #&y C r and x C M impliesy(x) #x. M -- M/r is a coveringspace if and only if r acts freelyand properlydiscontinuously on M. If M is strictlynoncompact, then the isotropysubgroups of I (M) are the maximal compact subgroups,and, if r acts properlydiscontinuously on M, it followsthat r acts freelyif and only if everyelement 7 I1 of r has infiniteorder. If r acts properlydiscon- tinuouslyon M, thenM/r is a llausdorfftopological space (althoughit need not be a manifold),and il/r is compactif and only if r is a uniformsub- group of I(M).

2.4. Holonomy groups. The homogeneousholonomy group H (M, x) of a Riemannianmanifold M at a point x C M is the group of linear trans- formationsof the tangentspaceMa, obtained by parallel translationof tangent- vectorsalong sectionallysmooth closed arcs based at x. The Riemannian metricgives M$,a positivedefinite inner product;H (M, x) is a subgroupof the correspondingorthogonal group and carriesthe induced topology. The restrictedhomogeneous holonomy group is the identitycomponent H0 (M, x), consistsof those elementsof H (M, x) obtained fromnullhomotopic closed arcs, and is a closed subgroupof the orthogonalgroup of Mx; in particular, Ho(M, x) is compact,and now H(M, x) is compactif and only if it has only finitelymany components. Thus we have a natural homomorphismof the fundamentalgroup 7ri(M, x) onto the quotientH (M, x) /H0(M, x). If M is connected,then we speak of H (M) and H, (M) in the same sense as 7r1(I). Suppose that M is a connectedsimply connected Riemannian symmetric DISCRETE GROUPS. 531 space, and M- M X M1 X * X Mt is the decompositioninto Euclidean and irreduciblenon-Euclidean parts. Then H (Me) = 1 and H (Mi, x) is the group of linear transformationsof (Mi) induced by the isotropysubgroup of I,(Mj) at x.

If M1and N are Riemannianmanifolds, x C M and y C N, then

H(M X N, (x, y)) ==H(M, x) X H(N, y).

3. Semisimplicityof discreteuniform subgroups. 3. 1. If a Cartan subgroupof a semisimpleLie group G is normalized by an elementg C G, then it is known [5, Proposition7. 7] that g is a semi- simpleelement of G. In orderto make the estimatesdescribed in ? 1, then, we need:

3.2. THEOREM.4 If r is a discreteuniform subgroup of a reductive Lie group G such that G/GOhas no elementof infiniteorder, then every elementof r is a semisimpleelement of G. The essentialpart of the reductionto the semisimplecase is given by: 3. 3. LEMMIA. Let r be a discreteuniform subgroup of a connected reductiveLie group G, let A be the connectedcenter of G, let G' be the semi- simple part of G, and let r'={g'E G': g'=- ay for some aC A, yC r}. Then r' is a discreteuniform subgroup of G' and r n A is a discreteuniform subgroupof A.

Proof. It is sufficientto considerthe case where Uo has no compact factor,for replacing G' by G'/K, whereK is the maximal compactnormal subgroupof G', affectsneither hypotheses nor conclusionsof the Lemma. Similarly,we may assume G- A X G'. Let {yi'} -> 1 be a sequence in r'; this gives us a sequence {yi} in r withyi = aryi',at C A. A beingcentral in G, { [ym,y]} = { [y/,y] } -> 1 for every-y C r; thus yi commnuteswith y forlarge i because r is discrete.r being finitelygenerated, it followsthat yi is central in r for large i. Now let 7r: G-> G' be the projection; r'= w7r(r). Being uniformin G, r has the Selbergdensity property (S) in G ([8, Lemma 1] or [4, Lemma 1. 4]); thus r' has the property(S) in G' [4, ? 1. 2]; it followsthat the centralizerof r' in G' is just the centerdf G' [4, Corollary4. 4], whenceyl' is centralin G'

4As the proof will show, the essential case is that of a semisimple linear group, for which the result is known; see Borel and Harish-Chandra, "Arithmetic subgroups of algebraic groups," Annals of Mathematics, vol. 75 (1962), pp. 485-535, esp. ? 11. 2. 532 JOSEPH A. WOLF. for large i. As G' has discrete , this contradicts {y1'} > 1. Thus F' is a discretesubgroup of W'. Let F be a compactfundamental domain for the action of r on G by righttranslations. r' is a uniformsubgroup of G' because r(F) is compact and G'== r' r(F). Our proofthat r n A be uniformin A is a modificationof an argument of A. Weil.5 Let a C A and retain the notation above. Then a = yf for somef C F and somey C r suchthat 7r(y) C 7r(F)-1nf r(r). vr(F)t' is compact and 7r(r) was just seen discrete; this gives {y<, * *,yt} C r such that the 7r(yi) exhaust7r(F)1 n7r(r);l it followsthat y =87yi for some i and some t t a CFr nA. In otherwords, a C (r n A). U yF for everya C A. As U yF is _=1 j=1 compact,it followsthat A/ (r n A) is compact. q. e. d. The semisimplecase is reducedto the linear semisimplecase by meansof: 3.4. LEM1A. Let r be a discreteuniform subgroup of a connected semisimple Lie group G, let Z be the center of G, and let 7,: G- G/Z be the projection. Then 7r(r) is a discreteuniform subgroup of G/Z and r has finiteindex in rF Z. Proof. It sufficesto show r -Z discretein G, and for this we may assume that G has no compactfactor. Let {yizi} -e 1 be a sequencein r z with y CEr and z ECZ. Given yC r, { [y y]} { [iz, y]} ->1; thus yi is centralin r for large i because r is discreteand finitelygenerated. As in the previouslemma, it followsthat y*C Z for large i, whenceyizi C Z for i large. Z being discrete,this contradicts{yiz} -> 1. q. e. d. 3. 5. Proof of Theorem3. 2. rn Go is a discreteuniform subgroup of Go, Go satisfiesthe hypothesesof the Theorem,every element of r has a finite powerin r n Go, and ymcannot be semisimpleunless y is semisimple. Thus we may assume G connected. Semisimplicityof y dependingonly on the automorphismad(y) of S, Lemmas 3.3 and 3.4 now allow us to assume that G is a centerlesssemisimple Lie group. Finally, everyautomorphism of a compactsimple Lie algebra being semisimple(because it preservesthe ,which is negativedefinite), we may factor G by its maximal compactnormal subgroupand assume that G has no compactfactor. In summary,we need onlyconsider the case whereG is a productof noncompact centerlessconnected simple Lie groups.

5 See his paper ";Discrete subgroups of Lie groups II," Annals of Mathematios, vol. 75. Observer that the argument proves: If A is a discrete uniform subgroup of a con- nected group S and P is a closed of S, then A n T is uniform in P if and only if the image of A in SI/ is discrete. DISCRETE GROUPS. 533

Let K be a maximalcompact subgroup of G and let 3 be the Riemannian symmetricspace G/K; G is the identitycomponent of the full group of isometriesof M, whencer is representedfaithfully on 31 by ;the action of r on M is properlydiscontinuous because r is discretein G. The adjoint representationof G representsr faithfullyas a definitelygenerated real matrixgroup, so r has a subgroupof finiteindex with no element# 1 of finiteorder [8, Lemma 8]. We cut r down to this subgroup,and may thus assume that no element7 1 of r has a fixedpoint on M, so M-> l/r is a coveringof Riemannianmanifolds which is a local (a Rieman- nian covering). M/r is compactbecause G/r is compact. M beinga complete connectedsimply connected Riemannian manifold with everysectional curva- ture ? 0, it follows [6] that every y C r preservessome geodesic on 31. Theorem3.2 followsfrom:

3. 6. LEMMA. Let 31 be a strictlynon-Euclidean Riemannian symmetric space, let G =I (M), the full groupof isometriesof M, and let g C G. Then g is a semisimpleelement of G if and onlyif somepower gin, m 70 O,preserves a geodesicon M. If g preservesa geodesicor throughx C M and has no fixed point on r, theng =7ep = pk wherekC U, ck(x) = x, and p is a transvection along

Proof. Let g be semisimple. Then g normalizesa E of 05 [5, Theorem7.6]. Choose m1n> such that gmiCH=exp(S4) E ; this is possiblebecause G is semisimpleand G/GOis finite. There is an element xC 3Mand a Cartan decompositionS == A +-$ where &? is the Lie algebra of the isotropy subgroup K of G at x, such that q = (.~ f)n + (& n $) This holds for compact Go by conjugacy of maximal tori and because an involutiveautomorphism must conservea ; it is known for noncompactlinear simple Go ([12], p. 107); it now follows in our case. Now gm= lkp with kEHnK and p=exp(X) for some XE1n$; thus gin preservesthe geodesica = {exp (tX)x} on M. Let gmpreservTe a geodesic a- on M; we wish to show g semisimple,and it sufficesto show gm semisimple. Thus we may assume g to preserve(r. As preservescr, we now replaceg by g2 if g has a fixedpoint on a-. a being a totallygeodesic submanifold of M, g induces an isometryof of onto itself, and the possiblereplacement of g by g2 showsthat g: ot - Ut+a for some real numbera, wheret is arc length. Let x C c, say x = o7, and take the (S = R +$ at x; this gives X C $ with at = exp (tX) x. k = exp(- aX) g C K, and Ic commuteswith X because it preservesevery at, whencek commuteswith p = exp(aX). Now g =k lp = plc, Ikis semisimple 534 JOSEPH A. WOLF. because it lies in a compactgroup K, p is a transvectionalong a, and p is semisimplebecause it is representedon (Mby a positivedefinite matrix. Being the productof two commutingsemisimple elements, g is semisimple. q. e. d.

3. 7. COROLLARY. If r is a discreteuniform subgroup of a reductiveLie groupG such that G/Gohas no elementof infiniteorder, and if A is an abelian subgroupof r, then A normalizesa Cartan subgroupof G.

Proof. Theorem3. 2 says that ad (A) is an abelian group of semisimple automorphismsof (; thus ad (A) is diagonable on ( over the complex numbers,and it followsthat ad (A) has a finitelygenerated subgroup D such that everyD-invariant subspace of ( is ad(A)-invariant. Finitelygenerated and abelian,D is of type (MP) * (see [5, p. 404]); thus D leaves invariant a Cartansubalgebra ) of ( [5, Theorem7. 6]. ) is ad (A) -invariantby choice of D, whenceA normalizesthe Cartansubgroup exp(g) of G. q. e. d.

3. 8. Remark. If j is a subgroupof r whichhas normalsubgroups *F such that * =- iJ D k1 D* D i1 D *o =1 with *j1/4t cyclic, then ad(F) is a group of semisimple of type (MP) * of (, so j normalizesa Cartan subgroupof G by [5, Theorem7. 6].

4. Bounds for reductive groups. 4. 1. Let G be a reductiveLie group. G has only a finitenumber of conjugacy classes of Cartan subgroups; let {H1, * * *, H-Jf} be a maximal collectionof mutuallynonconjugate Cartan subgroupsof G. Hi being a connectedabelian Lie group of dimensionr = rank.G, it is isomorphicto the productof a vectorgroup RTaiwith a torus Tr-at. The vectorrank of G, denotedv-rank. G, is definedto be the maximumof the ui If G/GOis finite, then each Hi has finiteindex in its normalizerin G; in this case thereis a smallest integer,which we defineto be the torsionrank of G and denote t-rank,G such that everyfinite abelian subgroup of G (which will auto maticallynormalize a Cartansubgroup by [5, Theorem7. 6]) can be expressed as the productof _ t-rank.G cyclicgroups.

4. 2. THEOREM. Let r be a discreteuniform subgroup of a reductive Lie group G such that G/Gois finite,and let A be an abelian subgroupof r. Then A can be expressedas the productof _ t-rank.G finitecyclic groups ilvitha free abelian group on -< v-rank.G generators; in particular,A is finitelygenerated. Furthermore,the second bound is bestpossible in the sense thatr has a subgroupwhich is freeabelian on v-rank.G generators. Finally, if X is a subgroupof r whichis the productof a finiteabelian group and a DISCRETE GROUPS. 535 freeabelian groupon m generators,if H is a Cartansubgroup of G normalized by X, and if r does not have an abelian subgroup I such that fn ' be of finiteindex in X but of infiniteindex in I, then Y n H is uniformin H.

Proof. By Corollary3. 7, A normalizes a Cartan subgroupA of G. A n A is finitelygenerated because it is a discretesubgroup of the connectedabelian Lie groupA, and A/n A has finiteindex in A becauseA has finiteindex in its normalizerin G; thus A is finitelygenerated. The firststatement now follows fromthe structuretheorem for finitelygenerated abelian groups and the definitionsof t-rank.G and v-rank.G. We need some lemmas for the other statements.

4.3. LEMMA.6 Let r be a discreteuniform subgroup of a Hausdorff topologicalgroup G, let D be a finitelygenerated subgroup of r, and let GD and rD be the respectivecentralizers of D in G and r. Then rD is a discrete uniformsubgroup of GD.

Proof. Let 7r: G--> G/r be the projection. As rD r n GD and G/r is a compactHausdorff space, rD is uniformin GD if and only if 7r(GD) is closed in G/r. If 7r(GD) is not closed in G/r, then 7r%,r(GD) is not closed in G, so there is a sequence {yi} in r of elementsdistinct mod GD and a sequence {gi} in GD with {yigi} e-> x for some x C G. Given d C D, { [yi, d] } = { [yigi,d] } -> [x, d], whence [yi, d] - [x, d] for large i because r is discrete,implying that y-lyj commutewith d for large i and j. As D is finitelygenerated, y-yj E GD for large i and j, contradictingour choice of the sequence {yi}. q. e. d. 4. 4. We will provethe last statementof the theorem,retaining the nota- tion of Lemma 4.3. Theorem 3.2 and an induction on the numberof generatorsof Y showthat Gz is a reductiveLie group,so its Lie algebrais a directsum f 0 (E 0) XTwith W abelian,6 a sum of compactsimple Lie algebras, and X a sum of noncompactsimple Lie algebras. By Lemma 3. 3 and assumptionon X, we see that % ==0; thus (Gz)o=TVXK where V is a vector group in exp(s) and K is a containingexp (C). nl H havingfinite index in X, and H C (Gz) O,gives m ? v-rank.H ? dim.V. On the otherhand, v n ry,is free abelian on dim.V generatorsby Lemmas 4.3 and 3.3; thus dim.V < m by assumption on .. It follows that m = v-rank.H, provingthe last statementof the Theorem. 4.5. LEMMA. Let H be a Cartan subgroupof a reductiveLie group G,

I This is essentially the same as A. Selberg's result [8, Lemma 2]. 536 JOSEPH A. WOLF. and suppose that H has an elementh such that,given g E G0, the number-of distinctabsolute values among the eigenvaluesof ad(h) (acting on () is at least as large as the numberof distinctabsolute values among the eigenvalues of ad(g). Then v-rank.H = v-rank.G. For ) has the maximalnumber of linearlyindependent real-valued roots among all Cartan subalgebrasof S. 4. 6.7 We will finishthe proof of Theorem4.2 by provingthe second statement. Let g C G,, the numberof whoseabsolute values of eigenvaluesin the adjoint representationis maximal among the elementsof Go. Taking powersonly separatesfurther the absolute values of the eigenvalues,so we have a neighborhoodU of 1 in Go and an integerm such that:

1. If ,u is the Hlaar measureon G/r and 7r: G--> G/r is the projection, then m > IA(G/r)/4-t(7r(U) ).

2. If 1 ? a _< m and ui C U, thenthe numberof distinctabsolute values of eigenvaluesof ad (u,gau2,l) is not less than the numberfor ad (g)v

We choose [8, Lemma 1] the integera and the uiC U such that u,gau,2-l = y C r, replace y by a powerwhich lies in a Cartan subgroupH of G, and observethat v-rank.H = v-rank.G by Lemma 4. 5. We may take -yto be a regular elementof G, whenceH/ (H n r) is compactby Lemma 4. 3 with D = {y}. Thus HIn r has a subgroupwhich is free abelian on v rank.G generators. The Theoremis proved. q. e. d.

5. Bounds for groups with Euclidean factor. Our tool forthe treat- ment of groups with Euclidean factor is the followinggeneralization of theoremsof L. Bieberbach[3] and L. Auslander [1]:

5. 1. THEOREM. Let K be a compact group of automorphismsof a connectedsimply connected nilpotent Lie group N, let E be the semidirect productKX N, let L E X G where G is a reductiveLie group with G/G, finite,and let r be a discreteuniform subgroup of L. Then r n (N X G) is a normalsubgroup of finiteindex in r whichis a discreteuniform subgroup of NX G. Proof.8 As in ? 3. 5, we may assume G to be connectedand without

7 Compare with paragraph 3, p. 151, of [8], where A. Selberg considers the case G=SL(n; R). 8This Theorem can be proved as a consequence of [2, Theorem 1], to which it is similar. We find it more convenient,however, to give an argument which is a variation on the proof of that result. DISCRETE GROUPS. 537

compactnormal subgroup; then G=A G' where A, the connectedcenter of G, is a vectorgroup and G', the semisimplepart of G, is withoutcompact factor. Let p: A X G'-* G be the projection;then (1 X p)-1 (r) is a discrete uniformsubgroup of E X A X Go with the same projectionon K as has r. ReplacingN by N X A, we see that it sufficesto provethe Theoremwhen G is semisimpleand withoutcompact factor. Let J be the closureof r* N in L, and let a: L-> G and /3:L ->K be be the projections. JOis solvableby the generalizedZassenhaus Lemma [2, Proposition2], and J is clearlynormalized by r; as a(r) has the Selberg densityproperty (S) in G (for r has it in L), oa(r)normalizesx(J) and G is semisimplewithout compact factor, it follows[4, Theorem4. 1] that a (J) has discreteclosure in G. Thus a (r) is discrete,being containedin a (J) The proofof the last part of Lemma 3. 3 now showsr n E uniformin E. r n E beinga discreteuniform subgroup of E, the generalizedBieberbach Theorem[1, Theorem1] showsthat (r n E) n N = r n N is a discreteuni- formsubgroup of N. r n N is normalin r, and is thus normalizedby /3(r). On the otherhand, interpretingK as a group of automorphismsof N, uni- formityof r n N in N impliesthat an elementof K is determinedby its action on r n N [1, Theorem2]. It followsthat ,8(r) is finite. q. e. d. 5. 2. COROLLARY. Let r be a discreteuniform subgroup of L =- E X G whereE is a semidirectproduct KR VP,K is a compactgroup of automorphisms of thevector group V, and G is a reductiveLie group with G/IG finite. If A is an abelian subgroupof r, thenA normalizessome Cartan subgroupH of G, and A can be expressedas the product of : t-rank.(K X G) finitecyclic groupswith a free abelian group on _ dim.V + v-rank.G generators. If r has no abelian subgroupX withthe propertythat A nA has finiteindex in A and infiniteindex in X, then A n (v X H) is uniformin V X HI. Finally, r has a subgroupwhich is free abelian on dim.V + v-rank.G generators. Proof. By Theorems3. 2 and 5. 1, everyelement of r is a semisimple elementof the reductiveLie group G" = rP (v X G), and G/IG " is finite. The Cartan subgroupsof G,"being of the formV X (Cartan subgroupof G), we have v-rank.G" = dim.V + v-rank.G. Every finitesubgroup of E being conjugateto a subgroupof K, we have t-rank.U" t-rank.(K X G). The Corollarynow followsfrom Theorem 4. 2. q. e. d.

6. Application to symmetricspaces. 6. 1. Let M be a connectedsimply connectedRiemannian symmetric space,and let L = I (M1),the full groupof isometriesof M. If M-= Mo X M' 538 JOSEPH A. WOLF. is the decompositionof 11Minto the productof its Euclidean and non-Euclidean parts,then L = E X G whereG = 1 (M1') is a semisimple(and thusreductive)

Lie group with G/Go finiteand E = I (M0) is the Euclidean group E (n), n = dim.M0, semidirectproduct 0 (n) 1Vwhere 0 (n) is the orthogonalgroup of a vectorspaceV which can be identifiedwith M, As everyflat maximal connectedtotally geodesic submanifold of M' is an orbitof a Cartan subgroup of maximalvector rank in G, we have v-rank.M' = v-rank.G. Thus v-rank.M dim.10 + v-rank.I(M').

6.2. THEOREM. Let M be a connectedsimply connectedRiemannian symmetricspace, let M0 and M' be the Euclidean and non-Euclideanparts of M., let r be a properlydiscontinuous group of isometriesof M with I/F compact,and let A be an abelian subgroupof r. Then M has a closed con- nectedA-invariant flat totallygeodesic submanifold SA whoseimage in Ml/r is compact,and A can be expressedas the productof C31t-rallk. (O (dim. M,) X I(M')) finitecyclic groups with a free abelian group on ?< v-rank.11 generators. If r has no abelian subgroupX withthe propertythat A fn has finiteindex in A butinfinite index in X, thenS%/AI is compact. Finally,r has a subgroupv whichis free abelian on v-rank.M generators,and St can be taken to be a maximalconnected flat totallygeodesic submanifold of M.

Proof. r acts by isometries,M/r is compact,and I (M) acts transitively on M withcompact isotropy subgroups; it followsthat r is a discreteuniform subgroupof I(M).

6. 3. Retainingthe notationof ? 6. 1, we will defineS, to be an orbit of one of the groups V X H where H is a Cartan subgroup of I (M') normalizedby A; such groups exist by Corollary5. 2. When the choice of H is made,we will choosex E M,,to be the originof V and choosey E M' such that ) = (, nl A) + ( & n $) where K is the isotropysubgroup of I (M') at y and v is the orthogonalcomplement of A in Z (M') under the Killing formof (M'). Then SA = (V XH) (x,y) =M, X H(y) is clearlyclosed, connected,flat and totally geodesic in M. Let 7r: I(M)-> I(M') be the projectionand let 8 E A. 7r(A) normalizesH, and is thus containedin K H; now ir(8) =kh. This gives 7(S)H(y) ==khH(y) ~=kHk-1(y) -=H(y), provingSA to be A-invariant. We choose H to be a Cartan subgroupof I (M') which is normalized by some maximal abelian subgroup 1i of r which contains A, and set =' 1 n (V X H); -' has finiteindex in 1 by Theorem5. 1 and finiteness of G/GO. Jcis finitelygenerated, and thus,by Lemma 4. 3, is uniformin its DISCRETE GROUPS. 539 centralizerA in I(M)); it followsthat 1' is uniformin V X H. Thus the image of SA in M/r is compact. As the bounds on the size of A followfrom Corollary5. 2, this provesthe firststatement of the Theorem. 6. 4. Observethat not only is the image of SA in M/r compact,but SA/I is compact. Thus SA/A is compactif A has finiteindex in (. The structureof abelian subgroupsof r is such that A has finiteindex in -DIif r has no abelian subgroupX withthe propertythat A n > has finiteindex in A but infiniteindex in E. This provesthe second statementof the Theorem. Corollary5. 2 shows that r has a subgroupwhich is free abelian on v-rank.M generators. If A is such a subgroup,then H is a Cartan subgroup of maximalvector rank in I (M'). Examiningthe compactand noncompact factorsof M' separately,we see that, for proper choice of y 3', SA is a maximal connectedflat totally geodesic submanifold of M. q. e. d. 6. 5. The completeconnected locally symmetricRiemannian manifolds are preciselythose manifoldswhose universal Riemannian covering manifold is symmetric.Thus Theorem6. 2 gives us: 6. 6. COROLLARY. Let N be a compact connected locally symmetric Riemannianmanifold, let D be an abelian subgroupof the fundamentalgroup ,rT(N), and let M, be the productof the compactirreducible factors of the universal Riemannian coveringmanifold M of N. Then N has a closed connectedflat totallygeodesic submanifold SD and an elementx C SD such that D is representedby closed geodesic arcs in SD based at x, and D can be expressedas the produtctof ? t-rank.I(M,) finitecyclic groups with a free abelian group on ? v-rank.31 generators;thus D is free abelian on ? v-rank.Mgenterators if M is strictlynoncompact. 7r1(N) has a subgroup P whichis freeabelian on v-rank.M generators,and Sp can be taken to be a maximal connectedflat totally geodesic submanifoldof N; thus N has a maximal connectedflat totallygeodesic submnanifoldwhich is closed in N. 7. Holonomy groups of locally symmetricspaces. Corollary7. 2 is the only part of ? 7 which uses the results of precedingsections; in fact, it uses only Theorem5. 1, whichdoes not dependon precedingresults.

7. 1. THEOREM. Let r be the group of deck transformationsof the universalRiemannian covering7r: M-- N of a completelocally symmetric Riemannianmanifold N, let M= Mo x M' be the decompositioninto Eu- clidean and non-Euclideanparts, and let V be the groupof pure translations of Mo. Then thereis a canonical isomorphismbetween

rp (V X 1 (M'))/(V X O(MH')) 540 JOSEPH A. WOLF. and the group H(N)/HO(N) of componentsof the homogeneousholonomy group of N.

Proof. Recall the homomorphism,8 of r ontoH(N,-r(x))/H0 (N, 7r (x)) definedby /8(y)= ty,HO(N, 7r(x)), where ty is the operation definedby ir (iry) and ry is any sectionally smooth arc in M from x to y (x). We can representty on the tangentspaceM,a as the differentialy*: Ma -My followedby parallel translationof tangentvectorsbackwards along ry. Let x E M = Mo X M' have repesentationx = (x0,x'), let K' be the isotropysubgroup of I (M') at x', and let P' = exp($') where $' is the orthogonalcomplement of ' in Z (M') under the Killing form of ' (M'). The everyelement of I (M') has expressionp'k' with kc'E K' and p' E P'. Observethat the identitycomponent Ko' is the isotropysubgroup of 1o (M') at x', and its actionon thetangentspace M., is thatof H (M, x) = Ho (N, 7r(x)); also, P' is the set of transvectionsalong geodesicsin M' whichpass throughx', and p*': Mm,'-- Mp'(w,)'is parallel translationalong the geodesic arc from x' to p'(x') on whichp' is a transvection. Let -yEr; -y yoy'with yoEI(Mo) and yEEI(M'). y'p'k' as above and yo pok0 with ko(xo) = xo and po E V; thus y = (pop') (kok'). Let y (yo,y') be the image of x; then po is transvectionalong a geodesic arc ro in Mo fromxo to yopp' is transvectionalong a geodesicarc I-' in M' from x' to y', we definery to be the geodesicarc in M fromx to y withprojections i' and ro and it is then clear that tyis representedby the differentialof kok' on M,. Thus the canonical homomorphism/8 of r onto H(N)/Ho (N) inducesthe isomorphismof the Theorem. q. e. d.

7.2. COROLLARY. Let N be a compactlocally symmetricRiemannian manifold. Then the homogeneousholonomy group H(N) is compact,i.e., H(N)/Ho(N) is finite. This followseasily fromTheorems 5. 1 and 7. 1.

7. 3. COROLLARY. Let M be a connectedsimply connected Riemannian symmetricspace with Euclidean part MO and non-Euclideanpart M'. If dim.MO> 2, or if dim.MO = 2 and M' is noncompact,then there are con- tinuummany affinely inequivalent diffeomorphic Rllemannian manifolds covered byM whichhave noncompacthomogeneous holonomy groups. If dim.Mo < 2, or if dim.MO = 2 and M' is compact,then every Riemannian manifold covered by M has compacthomogeneous holonomy group. If dim.MO = q -1 <2 and r is the orderof I(M')/IO(M'), then the numberof comn'ponentsof the homogeneousholonomy group of a Riemannianmanifold covered by M is a divisorof qr. DISCRETE GROUPS. 541

Proof.9 The last statementfollows from Theorem 7. 1 and the fact the groupof translationsof Mo has finiteindex q in I(Mo). This also provesthe secondstatement except when dim.Mo = 2. Let dim.MO = 2, let M' be com- pact,and let r be the group of decktransformations of a Riemanniancovering M-> N. I (M') is compact,whence the projectionro of r on I(Mo) is discrete. We wishto showthat the groupH of rotationparts of elementsof ro is finite. Let U be the linear subspaceof the vectorspaceMo whichis spannedby the translationparts of elementsof ro. If dim.U - 2, thenfiniteness of H follows fromthe Bieberbach theorem [3] (or fromTheorem 5. 1). If dim.U 1, then MOhas an orthonormalbasis {u, v} whereu spans U. As H normalizesU, everyelement of H hasmatrix ( ? ?) in thisbasis; thus H is finite.If

dim. U =0, then rO= H lies in a compactgroup, and thus finitebecause ro is discrete. Now H is finitein any case, and the last part of the secondstate- ment followsfrom Theorem 7. 1.

7.4. For each real numbert, we define9tg = ( (27rt) cS (2rt)/' the rotationwith eigenvaluesexp(? 27r/- it). Now suppose dim.MO> 2, viewMo as a vectorspace,and let {v1, . . ., v,} be an orthonormalbasis of Mo; 0 let At be the linear transformation(9t of Mo. If dim.Mo > 2, then 0 Ir_/ let yt be the isometry (no, nm)-- (Atmo + v3,m') of M = Mo X M; if dim.Mo =2 and M' is noncompact,then we have a tranvectionr in a non- compact irreduciblefactor of M', and we define yt to be the isometry (ino, i) .-- (Atmo,T'm') of M. In eithercase, yt generatesan infinitecyclic subgrouprt of l(M) which acts freelyand properlydiscontinuously on M. Thus Nt = M/rt is a iRiemannianmanifold covered by M. In both cases yt /ctatwhere 8t is a transvectionof M along some geodesica throughour basepointx, and whereAt is an isometryof M with at(x) = x. The element of H(Nt)/Ho(Nt) determinedby yt being representedon the tangentspace MN,by the differentialof yt follows by parallel translationalong a- from fAt(x) =- yt(x) to x, this elementis representedon MN,by the differential of At. By construction,this differentialis given by At on (MN), and is the identityon (M'),,; thus we may view the linear transformationAt as a generatorof H(Nt)/1H0(Nt). In particular,Nt has compacthomogeneous holonomygroup if and only if At has finiteorder, i.e., if and only if t is

9 As the proof will show, the essential case in when M is irreducible. This was originally handled by a lemma developed in discussions with H. C. Wang; in the present context, however, it is easier to appeal to Theorem 7. 1. 542 JOSEPH A. WOLF. rational. Affineequivalence induces isomorphismof holonomuygroups as groups of linear transformations;thus the firststatement follows from the fact that the Nt are mutuallyreal-analytically homeomorphic and we can choose continuummany algebraicallyindependent irrational numbers t. q. e. d. THE UNIVERSITY OF CALIFORNIA AT BERKELEY, THE INSTITUTE FOR ADVANCED STUDY.

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