Proceedings of the NationalAcademyof Science8 Vol.67,No. 2, p. 509, October 1970
On the Fundamental Group of Manifolds of Non-positive Curvature
Shing Tung Yau
UNIVERSITY OF CALIFORNIA, BERKELEY, CALIFORNIA94720
Communicatedby S. S.Chen, July17, 1970
Abstract. Weprove theorems on the structure of the fundamental group of a compact riemannian manifold of non-positive curvature. In particular, a con- jectureof J. Wolf [J. Differential Geometry, 2, 421-446 (1968) ] isproved.
Let M be a compact riemannian manifold of non-positive curvature and IIH (M) its fundamental group. We wish to announce the following theorems;
proofswill be found in athesis underpreparation.
THEOREM 1. Let G be a solvable subgroup of the fundamental group
H1,
(Al).ThenG is
offinite
indexover anabeliansubgroup.
Inparticular, wegive an affirmative answer to a problem raised by J. Wolf inthefollowing corollary.
COROLLARY 1. IfHI (M)issolvable,then M is
flat.
COROLLARY2. Everysolvable subgroup ofllH isfinitely generated.
THEOREM2. Let G be a subgroup of HI (M). IfG contains a subnormal maximal abeliansubgroupA,thenG isoffinite indexoverA.
COROLLARY 1. Suppose M is of strictly
neg#tive
curvature. If a subgroup G ofHI(M) is subnormal over an abeliansubgroip,
then G iscyclic. Hence, every solvablesubgroup ofHI (M)iscyclic.Iwishtoexpress mydeep gratitudetoProfessor B.Lawsonfor his manyinvaluablesugges- tions. I also thank ProfessorsChernandKobayashifor severalhelpfuldiscussions.
Afterthis paper waswritten,Professor J. Wolf informed me thatheandProfessor D. Gro- mollhavealsoobtainedaproof ofhis conjecture.
1Wolf, J. A., "Growth offinitelygenerated solvable groups and curvature of riemannian manifolds,"J.Differential Geometry,2, 421-446 (1968).
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