Lie groups that are closed at infinity

Abstract

A noncompact Riemannian manifold M 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 M . The isoperimetric constant of M M is defined by h ( M ) = inf { vol ( ∂ S ) / vol ( S ) } h(M) = \inf \{ {\text {vol}}(\partial S)/{\text {vol}}(S)\} where S S ranges over compact domains with boundary in M M . It is shown that a Lie group G G with left invariant metric is closed at infinity if and only if h ( G ) = 0 h(G) = 0 if and only if G G is amenable and unimodular. This result relates these geometric invariants of G G to the algebraic structure of G G since the conditions amenable and unimodular have algebraic characterizations for Lie groups. G G is amenable if and only if G G is a compact extension of a solvable group and G G is unimodular if and only if Tr ⁡ ( ad X ) = 0 \operatorname {Tr}({\text {ad}}\,X) = 0 for all X X in the Lie algebra of G 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.