Residually free groups do not admit a uniform polynomial isoperimetric function

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 $G_r\le F_2^{(1)}\times \cdots \times F_2^{(r)}$ 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.

9 pages, v2: Improvements to the exposition and minor corrections. Final accepted version, to appear in the Proceedings of the American Mathematical Society