On the Dehn functions of Kähler groups

We address the problem of which functions can arise as Dehn functions of Kähler groups. We explain why there are examples of Kähler groups with linear, quadratic, and exponential Dehn function. We then proceed to show that there is an example of a Kähler group which has Dehn function bounded below by a cubic function and above by n⁶. As a consequence we obtain that for a compact Kähler manifold having non-positive holomorphic bisectional curvature does not imply having quadratic Dehn function.