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Is $\mathbb{H}^n$ quasi-isometric to a leaf of a codimension 1 foliation of a compact manifold?
Let me add a comment. One knows that a codimension-one foliation without leafwise holonomy must be given by a Z^n action, by a result of Sacksteder
Sacksteder, Richard
Foliations and pseudogroups.
Am …