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

DOI:
https://doi.org/10.1090/S0002-9947-1989-0935533-2

MathSciNet review:
935533

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Abstract: A noncompact Riemannian manifold 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 . The isoperimetric constant of is defined by where ranges over compact domains with boundary in . It is shown that a Lie group with left invariant metric is closed at infinity if and only if if and only if is amenable and unimodular. This result relates these geometric invariants of to the algebraic structure of since the conditions amenable and unimodular have algebraic characterizations for Lie groups. is amenable if and only if is a compact extension of a solvable group and is unimodular if and only if for all in the Lie algebra of . 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1989-0935533-2

Keywords:
Lie group,
left invariant metric,
isoperimetric constant,
closed at infinity,
amenable,
unimodular

Article copyright:
© Copyright 1989
American Mathematical Society