The exact shape of the set which has the best isoperimetry in the continuous Heisenberg is (from what I know) a difficult open problem. This brought to wonder what is known in the discrete case?
More precisely... Given a finitely generated group $G$, build up its Cayley graph using a (finite symmetric) generating set $S$. For $F \subset G$, let $\partial F$ be the set of edges between $F$ and $F^\mathsf{c}$. Say a set is $\epsilon$-optimal if among all sets with $\frac{|\partial F|}{|F|} \leq \epsilon$ its cardinality is minimal. The precise question is:
$\textbf{Question}$: Is there a set $S$ as above and a sequence $\epsilon_n \to 0$ for which, we know what are the $\epsilon_n$ optimal sets in the discrete Heisenberg group?
I would be also interested to know if there any amenable group (except $\mathbb{Z}^d$) where we know the answer to this question. For $\mathbb{Z}^d$ with the usual generating sets, the $\epsilon$-optimal sets are (I think) determined by the Loomis-Whitney inequality (i.e. they are close as possible to hypercubes).
As a side remark, when a group $G$ can be written as a semi-direct product $G_1 \rtimes G_2$ of amenable groups (like the Heisenberg group), one builds Folner sets (i.e. sets with a small ratio $\frac{|\partial F|}{|F|}$) by taking $F = F_1 \times F_2$ where $F_i$ are some Folner sets of $G_i$. So it seems (but I did not find any reference to that effect) that the optimal sets for the Heisenberg could be of this form.
