Higher Divergence Functions for Heisenberg Groups

TL;DR

This paper establishes bounds on fillings of cycles in Heisenberg and Jet-Groups, leading to the computation of higher divergence functions and extending previous geometric analysis results.

Contribution

It introduces a Filling Theorem for Heisenberg Groups and generalizes the results to Jet-Groups, providing new bounds on divergence functions in these spaces.

Findings

Bounded the volume of fillings for cycles in Heisenberg Groups

Computed higher divergence functions for Heisenberg and Jet-Groups

Extended geometric analysis techniques to new classes of groups

Abstract

We prove a Filling Theorem for the Heisenberg Groups $H^{2n+1}$: For a given $k$-cycle $a$ we construct a $(k+1)$-chain $b$ (the filling) with boundary $\partial b=a$ and controlled volume. For this filling $b$ we prove a uniform bound on the distance of points in $b$ to its boundary $a$. Using this we compute the higher divergence functions for the Heisenberg Groups $H^{2n+1}$. Further we generalise these results to the Jet-Groups $J^m(\mathbb R^n)$ for dimension less or equal $n$ .