Residually free groups do not admit a uniform polynomial isoperimetric function

TL;DR

This paper demonstrates that residually free groups cannot have a uniform polynomial isoperimetric function by constructing specific subgroups with Dehn functions growing faster than any fixed polynomial.

Contribution

It proves that no uniform polynomial isoperimetric function exists for residually free groups by analyzing subgroups of direct products of free groups with increasing Dehn functions.

Findings

Constructed subgroups with Dehn functions bounded below by n^r

Showed residually free groups lack a uniform polynomial isoperimetric function

Extended previous examples of non-coabelian subdirect products

Abstract

We show that there is no uniform polynomial isoperimetric function for finitely presented subgroups of direct products of free groups, by producing a sequence of subgroups $G_r\leq F_2^{(1)} \times \dots \times F_2^{(r)}$ of direct products of 2-generated free groups with Dehn functions bounded below by $n^{r}$. The groups $G_r$ are obtained from the examples of non-coabelian subdirect products of free groups constructed by Bridson, Howie, Miller and Short. As a consequence we obtain that residually free groups do not admit a uniform polynomial isoperimetric function.