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We show that every collapsed Gromov-Hausdorff limit of compact Heisenberg manifolds endowed with left-invariant Riemannian/sub-Riemannian metrics is isometric to a flat torus.We say that a sequence of sub-Riemannian manifolds collapses if their total measure with respect to Popp's volume converges to zero.
Kenshiro Tashiro (Wed,) studied this question.