Filling Invariants of Stratified Nilpotent Lie Groups

TL;DR

This paper investigates the growth of filling and divergence functions in stratified nilpotent Lie groups, revealing dimension-dependent behaviors and establishing algebraic criteria for their growth rates compared to Euclidean spaces.

Contribution

It introduces an algebraic condition on Lie algebras that determines the growth of filling functions in stratified nilpotent Lie groups and provides criteria for faster growth in specific dimensions.

Findings

Low-dimensional filling functions grow like Euclidean space

High-dimensional filling functions grow slower than Euclidean space

Bounded higher divergence functions for these groups

Abstract

Filling invariants are measurements of a metric space describing the behaviour of isoperimetric inequalities. In this article we examine filling functions and higher divergence functions. We prove for a class of stratified nilpotent Lie groups that in the low dimensions the filling functions grow as fast as the ones of the Euclidean space and in the high dimensions slower than the filling functions of the Euclidean space. We do this by developing a purely algebraic condition on the Lie algebra of a stratified nilpotent Lie group. Further, we find a sufficient criterion for such groups to have a filling function in a special dimension with faster growth as the appropriate filling function of the Euclidean space. Further we bound the higher divergence functions of stratified nilpotent Lie groups.