Dehn function for undistorted subgroups of a product of free groups 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?
 A: The answer is no: actually by Baumslag-Roseblade (1984: [BR]) either $G$ is commensurable to a product of free groups (hence has linear or quadratic Dehn function), or is not finitely presented (so the Dehn function is infinite, or not defined, as you wish). The latter case occurs if $H$ is an infinite word hyperbolic group and $f:F_m\to H$ is a non-bijective surjection and $H$ is the fibre product $$\{(g,h)\in F_m\times F_m:f(g)=f(h)\}.$$ 
Edit 1 (reply to Lee Mosher's comment, copy from a comment dated Sep 14 '12) In general the Dehn function of $H$ is equivalent to the distortion of $G$. This is not completely formal (there's a little Van Kampen diagram cuisine) but in the case of $H=\mathbf{Z}$ it's simple to verify by hand. I saw the general result written somewhere but I can't remember right now.
Edit 2 (Extracted from Robert Young's comment to his reply) There's a proof for [this non-distortion fact] mentioned in Olshanskii, Sapir [0S, Theorem 2].

References:
[BR] G. Baumslag, J. Roseblade. Subgroups of direct products of free groups. J. London Math. Soc. (2) 30 (1984), no. 1, 44-52. Journal link (restricted access). MR link
[OS] A. Ol'shanskii, M. Sapir. Length and area functions on groups and quasi-isometric Higman embeddings. ArXiv Link. Internat. J. Algebra Comput. 11 (2001), no. 2, 137-170. Journal link (restricted access). MR link
A: 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.
