A note on Freiman models in Heisenberg groups

Abstract: Green and Ruzsa recently proved that for any $s\ge2$, any small squaring set $A$ in a (multiplicative) abelian group, i.e. $|A\cdot A|<K|A|$, has a Freiman $s$-model: it means that there exists a group $G$ and a Freiman $s$-isomorphism from $A$ into $G$ such that $|G|<f(s,K)|A|$. In an unpublished note, Green proved that such a result does not necessarily hold in non abelian groups if $s\ge64$. The aim of this paper is improve Green's result by showing that it remains true under the weaker assumption $s\ge6$.