Combinatorial isoperimetric inequality for the free factor complex

We show that the free factor complex of the free group of rank greater than or equal to 4 does not satisfy a combinatorial isoperimetric inequality: that is, for every natural number N, there is a loop c_N of length 4 in the free factor complex such that the number of 2-simplices required to fill c_N grows at least as a linear function of N. To prove the result, we construct a coarsely Lipschitz function from the `upward link’ of a free factor to the set of integers.