On the Dehn functions of central products of nilpotent groups

Abstract

We determine the Dehn functions of central products of two families of filiform nilpotent Lie groups of arbitrary dimension with all simply connected nilpotent Lie groups with cyclic centre and strictly lower nilpotency class. We also determine the Dehn functions of all central products of nilpotent Lie groups of dimension at most $5$ with one-dimensional centre. This confirms a conjecture of Llosa Isenrich, Pallier and Tessera for these cases, providing further evidence that the Dehn functions of central products are often strictly lower than those of the factors. Our work generalises previous results of Llosa Isenrich, Pallier and Tessera and produces an uncountable family of nilpotent Lie groups without lattices whose Dehn functions are strictly lower than the ones of the associated Carnot-graded groups. A consequence of our main result is the existence of an infinite family of groups such that Cornulier's bounds on the $e$ for which there is an $O(r^e)$-bilipschitz equivalence between them and their Carnot-graded groups are asymptotically optimal, as the nilpotency class goes to infinity.