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On isoperimetric profiles of finitely generated groups. (English) Zbl 1049.20024

Let \(G\) be a finitely generated group equipped with a finite generating set \(S\) and with the corresponding word metric. The author uses the notion of the Følner function of \(G\) with respect to \(S\). We recall that for any subset \(V\subset G\), its boundary \(\partial V\) is defined as \[ \partial V=\{v\in G:d(v,G\setminus V)=1\}. \] It is known that since \(G\) is amenable, one can set, for any \(\varepsilon\geq 0\), \(F_{G,S}(\varepsilon)\) to be the minimum of integers \(k\) such that there exists \(V\subset G\) with \(\text{Card}(V)=k\) and \(\text{Card}(\partial V)/\text{Card}(V)<\varepsilon\). The author sets then, for any integer \(n\geq 0\), \[ \text{Føln}_{G,S}(n)=F_{G,S}(1/n). \] There are natural equivalence relations between functions such as \(F_{G,S}\) (respectively \(\text{Føln}_{G,S}\)) which behave well with respect to change of system of generators of \(G\), and the author denotes the equivalence classes of such functions as \(F_G\) (respectively \(\text{Føln}_G\)). The functions \(F_{G,S}\) and \(\text{Føln}_{G,S}(n)\) as well as their equivalence classes \(F_G\) and \(\text{Føln}_G\) are called Følner functions. The equivalence classes of functions \(F_G\) and \(\text{Føln}_G\) are quasi-isometry invariants of the group \(G\). Følner functions were first considered by Vershik, and they were interpreted by Varopoulos in terms of discrete kernel decay. Coulhon and Saloff-Coste proved that Følner functions of groups of exponential growth grow at least exponentially.
In the paper under review, the author describes the asymptotic growth of Følner functions for a class of examples, which include examples considered by Vershik, Pittet and Saloff-Coste, as well as new examples. She establishes a general connection between the growth of Følner sets (“isoperimetric profiles”) and certain other types of isoperimetric inequalities. She describes up to multiplicative constants the isoperimetric profiles of wreath products and of other related groups, and she provides the first examples where the asymptotic growth of the isoperimetric profile is neither polynomial nor exponential.

MSC:

20F65 Geometric group theory
20F05 Generators, relations, and presentations of groups
20E22 Extensions, wreath products, and other compositions of groups
43A07 Means on groups, semigroups, etc.; amenable groups
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C30 Differential geometry of homogeneous manifolds
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