*preprint*

**Inserted:** 6 aug 2020

**Last Updated:** 6 aug 2020

**Year:** 2020

**Abstract:**

For every $k\geqslant 3$, we exhibit a simply connected $k$-nilpotent Lie group $N_k$ whose Dehn function behaves like $n^k$, while the Dehn function of its associated Carnot graded group behaves like $n^{k+1}$. This property and its consequences allow us to reveal three new phenomena. First, since those groups have uniform lattices, this provides the first examples of pairs of finitely presented groups with bilipschitz asymptotic cones but with different Dehn functions. The second surprising feature of these groups is that for every even integer $k \geqslant 4$ the centralized Dehn function of $N_k$ behaves like $n^{k-1}$ and so has a different exponent than the Dehn function. This answers a question of Young. Finally, we turn our attention to sublinear bilipschitz equivalences (SBE), which are weakenings of quasiisometries where the additive error is replaced by a sublinearly growing function; they were introduced by Cornulier. The group $N_4$ had specifically been considered by Cornulier who suspected the existence of a positive lower bound on the set of $r>0$ such that there exists an $n^r$-SBE between $N_4$ and its associated Carnot graded group, strengthening the fact that those two groups are not quasiisometric. We confirm his intuition, thereby producing the first example of a pair of groups for which such a positive lower bound is known to exist. More generally, we show that $r_k = 1/(2k - 1)$ is a lower bound for the group $N_k$ for all $k\geqslant 4$.

**Tags:**
GeoMeG