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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Lie groups that are closed at infinity

Author: Harry F. Hoke
Journal: Trans. Amer. Math. Soc. 313 (1989), 721-735
MSC: Primary 58F18; Secondary 22E15
MathSciNet review: 935533
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Abstract: A noncompact Riemannian manifold $ M$ is said to be closed at infinity if no bounded volume form which is also bounded away from zero can be written as the exterior derivative of a bounded form on $ M$ . The isoperimetric constant of $ M$ is defined by $ h(M) = \inf \{ {\text{vol}}(\partial S)/{\text{vol}}(S)\} $ where $ S$ ranges over compact domains with boundary in $ M$. It is shown that a Lie group $ G$ with left invariant metric is closed at infinity if and only if $ h(G) = 0$ if and only if $ G$ is amenable and unimodular. This result relates these geometric invariants of $ G$ to the algebraic structure of $ G$ since the conditions amenable and unimodular have algebraic characterizations for Lie groups. $ G$ is amenable if and only if $ G$ is a compact extension of a solvable group and $ G$ is unimodular if and only if $ \operatorname{Tr}({\text{ad}}\,X) = 0$ for all $ X$ in the Lie algebra of $ G$. An application is the clarification of relationships between several conditions for the existence of transversal invariant measures for a foliation of a compact manifold by the orbits of a Lie group action.

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Keywords: Lie group, left invariant metric, isoperimetric constant, closed at infinity, amenable, unimodular
Article copyright: © Copyright 1989 American Mathematical Society