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If $X$ is a smooth manifold on which a Lie group $G$ acts properly and cocompactly (meaning $X/G$ is compact), then one can find a compactly support cut-off function $c:X\rightarrow\mathbb{R}$ such ...

Noncommutative geometry associates a $C^*$-algebra $C^*(S,{\cal F})$ with a foliation $\cal F$ on a manifold $S.$ Did somebody study this construction for noncompact surfaces $S$? What I am really ...