Cone‐equivalent nilpotent groups with different Dehn functions

Abstract

For every k ⩾ 3 $k\geqslant 3$ , we exhibit a simply connected k $k$ -nilpotent Lie group N k $N_k$ whose Dehn function behaves like n k $n^k$ , while the Dehn function of its associated Carnot graded group gr ( N k ) $\mathsf {gr}(N_k)$ behaves like n k + 1 $n^{k+1}$ . 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 bi-Lipschitz asymptotic cones but with different Dehn functions. The second surprising feature of these groups is that for every even integer k ⩾ 4 $k \geqslant 4$ , the centralised Dehn function of N k $N_k$ behaves like n k − 1 $n^{k-1}$ and so has a different exponent than the Dehn function. This answers a question of Young. Finally, we turn our attention to sublinear bi-Lipschitz equivalences (SBEs). Introduced by Cornulier, these are maps between metric spaces inducing bi-Lipschitz homeomorphisms between their asymptotic cones. These can be seen as weakenings of quasi-isometries where the additive error is replaced by a sublinearly growing function v $v$ . We show that a v $v$ -SBE between N k $N_k$ and gr ( N k ) $\mathsf {gr}(N_k)$ must satisfy v ( n ) ≽ n 1 / ( 2 k + 2 ) $v(n)\succcurlyeq n^{1/(2k + 2)}$ , strengthening the fact that those two groups are not quasi-isometric. This is the first instance where an explicit lower bound is provided for a pair of SBE groups.