Cone-equivalent nilpotent groups with different Dehn functions

TL;DR

This paper constructs specific nilpotent Lie groups with distinct Dehn function behaviors, revealing new phenomena in geometric group theory, including examples with bilipschitz asymptotic cones but different Dehn functions, and establishing lower bounds for sublinear bilipschitz equivalences.

Contribution

It introduces new examples of nilpotent groups with unique Dehn function properties and analyzes their asymptotic cone relationships and sublinear bilipschitz equivalences.

Findings

Groups have different Dehn functions despite bilipschitz asymptotic cones.

For even k, centralized Dehn function differs from Dehn function, answering a question of Young.

Established a lower bound for sublinear bilipschitz equivalences between specific groups.

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 $\mathsf{gr}(N_k)$ 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). Introduced by Cornulier, these are maps between metric spaces inducing bi-Lipschitz homeomorphisms between their asymptotic cones. These can be seen as weakenings of quasiisometries where the additive error is replaced by a sublinearly growing function $v$. We show that a $v$-SBE between $N_k$ and $\mathsf{gr}(N_k)$ must satisfy $v(n)\succcurlyeq n^{1/(2k + 2)}$, strengthening the fact that those two groups are not quasiisometric. This is the first instance where an explicit lower bound is provided for a pair of SBE groups.