Let $G$ be a finitely generated subgroup of a product of two finite rank free groups $F\_m \times F\_n$. If there is a Lipschitz retraction $F\_m \times F\_n \to G$ with respect to word metrics, then $G$ is undistorted in $F\_m \times F\_n$, and the Dehn function of $G$ has a quadratic upper bound.


Suppose now that we require only that $G$ is undistorted in $F\_m \times F\_n$. Is it still true that the Dehn function of $G$ has a quadratic upper bound?



Dehn function for undistorted subgroups of a product of free groups
The Bieri-Stallings subgroup of $F\_n\times F\_n$ is undistorted and finitely generated, but not finitely-presented, so in some sense it has an infinite Dehn function. It's the kernel of the map $F\_n\times F\_n\to \mathbb{Z}$ which sends each generator to 1, and it's generated by elements of the form $g\_ih\_j^{-1}$ where $g\_i$ and $h\_j$ are generators of the two different factors.


