Nilpotent groups without exactly polynomial Dehn function

TL;DR

This paper establishes that some finitely generated nilpotent groups have Dehn functions with growth rates that are not exactly polynomial, answering a longstanding open question in geometric group theory.

Contribution

It proves super-quadratic lower bounds for the filling area function of certain Carnot groups, demonstrating the existence of nilpotent groups with non-polynomial Dehn function growth.

Findings

Existence of nilpotent groups with non-polynomial Dehn functions

Super-quadratic lower bounds for filling area functions

Answer to a long-standing question about Dehn function growth rates

Abstract

We prove super-quadratic lower bounds for the growth of the filling area function of a certain class of Carnot groups. This class contains groups for which it is known that their Dehn function grows no faster than $n^2\log n$. We therefore obtain the existence of (finitely generated) nilpotent groups whose Dehn functions do not have exactly polynomial growth and we thus answer a well-known question about the possible growth rate of Dehn functions of nilpotent groups.