Background
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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 $A$.




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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 $d<\dim(G)<\infty$?
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Going a bit further:



> 
> Can the isoperimetric dimension of a $d$-generated group attain any value?
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Can the isoperimetric dimension of a d-generated group attain any value?
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](http://arxiv.org/abs/1204.1613) of Breuillard and Le Donne about volumes of spheres might suggest it can be greater if the nilpotency length is greater than 2.


