Generalized Dehn Functions II

TL;DR

This paper investigates the properties and relationships of isoperimetric profiles and Dehn functions across all dimensions for groups and manifolds, revealing their finiteness, uniqueness, and equivalence in higher dimensions.

Contribution

It proves the existence, finiteness, and uniqueness of isoperimetric profiles in all dimensions and shows their equivalence in dimensions 4 and higher, extending understanding of Dehn functions.

Findings

Dehn functions equal FV^{n+1} for n ≥ 3

Profiles coincide in dimensions 4 and higher

Decidable word problem influences growth rates

Abstract

We establish the existence, finiteness, and uniqueness up to scaling of various isoperimetric profiles of a group, in all dimensions. We also show that these profiles all coincide in dimensions 4 and higher; in particular, the nth Dehn function is equal to FV^{n+1} for n at least 3. Even for dimension 3, there is significant overlap. When a group has decidable word problem, this has mild consequences for the growth rates of its profiles. We also establish a metric analogue for highly connected Riemannian manifolds.