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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 $A$.

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?

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In the general context of finitely generated groups, there is little reason to relate $d$ with the number of generators: for instance you can reduce or increase the minimal number of generators by embedding $\mathbf{Z}^k$ as a finite index subgroup of a suitable virtually abelian group. – YCor Dec 28 '12 at 23:56
up vote 3 down vote accepted

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.

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Could you please expand on what makes you think that non-integer values might be possible? – R W Dec 30 '12 at 7:06
it depends. if the conjecture is that nilpotent groups have isoperimetric dimension equal to the degree of growth, then this is an integer. But if this this is not true, then there is no particular reason to expect that this is an integer. The upper bound given by Breuillard and Le Donne for balls in (torsion free f.g.) nilpotent groups of nilpotency length $>2$ is not always an integer. – YCor Dec 30 '12 at 12:15
Thank you - so the evidence so far is rather circumstantial. – R W Dec 31 '12 at 10:35
Thanks for this. It seems that seeking a connection between the isoperimetric dimension and number of generators is indeed rather wrongheaded. What I was ultimately asking about, I suppose, is the existence of some kind of "dimension gap." So the issues you raise--of whether being finite-dimensional is equivalent to having polynomial growth, and of whether the isoperimetric dimension must be an integer--are quite interesting. – Jan Jan 5 '13 at 20:44

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