Filling inequalities for nilpotent groups

TL;DR

This paper investigates filling inequalities in nilpotent groups, providing bounds on higher-order Dehn functions and filling invariants for specific Carnot groups, with implications for Gromov's conjecture and group constructions.

Contribution

It introduces new bounds on higher-order Dehn functions for Carnot groups, including Heisenberg and jet groups, and constructs groups with large nilpotency class but Euclidean filling volume functions.

Findings

Bounded higher-order Dehn functions for certain Carnot groups.

Constructed groups with large nilpotency class and Euclidean filling volume.

Proved part of Gromov's conjecture on higher-order filling functions.

Abstract

We bound the higher-order Dehn functions and other filling invariants of certain Carnot groups using approximation techniques. These groups include the higher-dimensional Heisenberg groups, jet groups, and central products of two-step nilpotent groups. Some consequences of this work are a construction of groups with arbitrarily large nilpotency class that have euclidean n-dimensional filling volume functions, and a proof of part of a conjecture of Gromov on the higher-order filling functions of the higher-dimensional Heisenberg groups.