Enlargeable Foliations and the Monodromy Groupoid: Infinite Covers

In this paper, we prove that the foliated Rosenberg index of a possibly noncompactly enlargeable, spin foliation is nonzero. It generalizes our previous result. The difficulty brought by the noncompactness is reflected in the infinite dimensionality of some vector bundles which, fortunately, can be reduced to finite dimensional vector bundles by the idea of relative index theorem and $KK$-equivalence between the $C^{\ast }$-algebra of compact operators and $\mathbb{C}$.