Discrete Groups, Symmetric Spaces, and Global Holonomy Author(s): Joseph A. Wolf Reviewed work(s): Source: American Journal of Mathematics, 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 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. 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 group of isometriessuch that M/r is compact. If everysectional curvature of M is negative,in particularif M is a noncompactirreducible symmetric space 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 Euclidean space 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 abelian group 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 Riemannian manifold 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 Euclidean group,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 Lie group G, Go will denote the identity component,(M will denotethe Lie algebra, and exp: S -> G will denotethe exponentialmap. G and (Mare called reductiveif the adjoint representation 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 open set 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.