Cone-equivalent nilpotent groups with different Dehn functions

For every (Formula presented.), we exhibit a simply connected (Formula presented.) -nilpotent Lie group (Formula presented.) whose Dehn function behaves like (Formula presented.), while the Dehn function of its associated Carnot graded group (Formula presented.) behaves like (Formula presented.). 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 (Formula presented.), the centralised Dehn function of (Formula presented.) behaves like (Formula presented.) 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 (Formula presented.). We show that a (Formula presented.) -SBE between (Formula presented.) and (Formula presented.) must satisfy (Formula presented.), 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.