Question

Background

The isoperimetric dimension of a finitely generated group $G$, which we denote by $\dim(G)$, is the largest number $d$ such that any Cayley graph $\Gamma$ of $G$ (constructed with respect to a finite generating set) satisfies a $d$-dimensional isoperimetric inequality, i.e. \begin{equation} |\partial A|\geq C|A|^{(d-1)/d} \end{equation} for all finite subsets $A\subseteq\Gamma$, where $C$ is some constant (which depends on $\Gamma$ and $d$ but not on $A$). Here $\partial A$, the bounday of $A$, is the set of vertices in $\Gamma\backslash A$ which have a neighbor in $A$.

---

Suppose now that $G$ is a $d$-generated group, i.e. a quotient of $\mathbb{F}_d$, the free group of rank $d$. Then provided $d>1$, $\dim(G)$ may attain any value in the set $\{0,\ldots,d\}\cup\{\infty\}$, as is evidenced, for instance, by the free Abelian groups $\mathbb{Z}^k$, where $0\leq k\leq d$ (since $\dim(\mathbb{Z}^k)=k$), and the free group $\mathbb{F}_d$ itself (since $\dim(\mathbb{F}_d)=\infty$). My question is:

What are examples of $d$-generated groups $G$ that satisfy $d<\dim(G)<\infty$?

Going a bit further:

Can the isoperimetric dimension of a $d$-generated group attain any value?

Answer 1

Denoting by $C_k$ a cyclic group of order $k$, the wreath product $\mathbf{Z}\wr C_k=\mathbf{Z}^k\rtimes C_k$ is 2-generated (hence $d$-generated for any $d\ge 2$) and has isoperimetric dimension (in the above sense) $k$.

It's likely that the "isoperimetric dimension" is finite only for f.g. groups with polynomial growth. In this case the computation is not easy and might (?) give rise to non-integral values. I do not know whether the terminology "$d$-dimensional isoperimetric inequality" is motivated by any example beyond the Euclidean setting. A natural question is whether it can be greater than the polynomial degree of growth. The results of Breuillard and Le Donne about volumes of spheres might suggest it can be greater if the nilpotency length is greater than 2.