An isoperimetric inequality for the Heisenberg groups

We show that the Heisenberg groups $ \cal {H}^{2n+1} $ of dimension five and higher, considered as Riemannian manifolds, satisfy a quadratic isoperimetric inequality. (This means that each loop of length L bounds a disk of area ~ L 2.) This implies several important results about isoperimetric inequalities for discrete groups that act either on $ \cal {H}^{2n+1} $ or on complex hyperbolic space, and provides interesting examples in geometric group theory. The proof consists of explicit construction of a disk spanning each loop in $ \cal {H}^{2n+1} $ .