# Compactness theorem for Heisenberg manifolds with Sub-Riemannian metrics under the Gromov--Hausdorff topology.

2020/06/30 Tue 15:00 - 16:30

We investigate a condition for a family of sub-Riemannian manifolds being compact under the Gromov--Hausdorff topology. In this talk, we introduce a volume form on the Heisenberg Lie group endowed with Sub-Riemannian metrics of corank $0$ or $1$ to give an analogy of Mahler's compactness theorem for compact Heisenberg manifolds, which is a quatient space of the Heisenberg Lie group by a uniform discrete subgroup. Namely we show that if a family of such Heisenberg manifolds has a lower bound of systole and an upper bound of total measure, then it is relatively compact.

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