An isoperimetric inequality for the Heisenberg groups

We show that the Heisenberg groups \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \cal {H}^{2n+1} $\end{document} 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 ~ L2.) This implies several important results about isoperimetric inequalities for discrete groups that act either on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \cal {H}^{2n+1} $\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \cal {H}^{2n+1} $\end{document}.