## C. Llosa Isenrich - G. Pallier - R. Tessera

# Cone-equivalent nilpotent groups with different Dehn functions

created by pallier on 06 Aug 2020

[

BibTeX]

*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