The Dehn function of a finitely presented group is an important asymptotic invariant. It provides a quantitative measure for the difficulty of detecting whether a word in its generators represents the trivial element in the group.

Asymptotic cones of groups capture the geometry of their Cayley graphs when "viewed from infinity". There are well-known connections between asymptotic cones and Dehn functions, raising the question if the Dehn function of a group can be determined from its asymptotic cones.

We will show that even for nilpotent groups -- a class of groups for which Dehn functions and asymptotic cones are known to be particularly well-behaved -- this is not possible, by exhibiting a family of finitely presented $k$-step nilpotent groups $N_k$ with Dehn function $n^k$, but such that their associated Carnot graded groups have Dehn function $n^{k+1}$.

The computation of the precise Dehn functions of the $N_k$ also reveals other interesting phenomena. For even $k \geq 4$ the centralized Dehn function of the groups $N_k$ is $n^{k-1}$ and thus its exponent differs from the one of the regular Dehn function.

This is joint work with Gabriel Pallier and Romain Tessera.