High-dimensional fillings in Heisenberg groups

TL;DR

This paper demonstrates that high-dimensional cycles in Heisenberg groups can be efficiently approximated by simplicial cycles, enabling the calculation of higher-order Dehn functions and confirming Gromov's conjecture.

Contribution

It introduces a method using intersections with horizontal manifolds to compute higher-order Dehn functions in Heisenberg groups, solving a longstanding conjecture.

Findings

Higher-order Dehn functions of Heisenberg groups are explicitly calculated.

Efficient approximation of high-dimensional cycles by simplicial cycles is established.

Gromov's conjecture on Dehn functions in Heisenberg groups is proved.

Abstract

We use intersections with horizontal manifolds to show that high-dimensional cycles in the Heisenberg group can be approximated efficiently by simplicial cycles. This lets us calculate all of the higher-order Dehn functions of the Heisenberg groups, thus proving a conjecture of Gromov.