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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.