Lie Groups That are Closed at Infinity

Harry F. Hoke
1989 Transactions of the American Mathematical Society  
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{vol(3.S)/ vol^)} where 5 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
more » ... is 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 Tr(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.
doi:10.2307/2001426 fatcat:gkh7mjilpzha7ihvi3q2kt7ipu