Lie groups that are closed at infinity

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.