Abstract.
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 \( \cal {H}^{2n+1} \).
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Submitted: April 1997, Final version: November 1997
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Allcock, D. An isoperimetric inequality for the Heisenberg groups. GAFA, Geom. funct. anal. 8, 219–233 (1998). https://doi.org/10.1007/s000390050053
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DOI: https://doi.org/10.1007/s000390050053