It's well know that the first order Dehn function of $F$ is quadratic. Is a similar result known for its second-order, or even higher-order, Dehn function?
The second-order Dehn function of a group $G=G(\mathcal{P})$ is defined as follows. For a generating set $X$ of the second-homotopy module $\pi_2(\mathcal{P})$ the volume of $\xi \in \pi_2(\mathcal{P})$ is defined to be the minimum number of elements of $X$ required to generate $\xi$. The second-order Dehn function is then given by $$\delta(n)=\mbox{max}\{volume(\xi):area(\xi)\leq n\}.$$ This is a natural extension of the notion "max area over min length" as in the first-order case.
