Isoperimetry in finitely generated groups (Q6610774)

The main result of the book chapter under review is a sharpening of the isoperimetric inequalities for finite sets in the Cayley graph of a finitely generated group previously obtained by \textit{T. Coulhon} and \textit{L. Saloff-Coste}, Rev. Mat. Iberoam. 9, No. 2, 293--314 (1993; Zbl 0782.53066) and \textit{M. Gromov} et al., Metric structures for Riemannian and non-Riemannian spaces. Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes. Edited by J. LaFontaine and P. Pansu. Boston, MA: Birkhäuser (1999; Zbl 0953.53002). To do this, the authors introduce a function transform \(f \mapsto \mathcal{U}_f\) which, as they show, is related to the Legendre transform as follows: if \(f, g \colon \mathbb{R}_{>0} \rightarrow \mathbb{R}\) are functions such that \(f(x)g(\frac{1}{x}) = x\), then \(\mathcal{U}_g(t) = t \mathcal{L}_f(\frac{1}{t})\).\N\NLet \(\Gamma\) be a group generated by a finite set \(S \subseteq \Gamma\). The isoperimetric inequality obtained by the authors then takes the form\N\[\N\frac{|\partial_S D|}{|D|} \geqslant \mathcal{U}_{\gamma}(|D|)\N\]\Nfor all finite sets \(D \subseteq \Gamma\), where \(\gamma\) is the growth function of \(\Gamma\) and \(\partial_S D\) is the \emph{inner boundary} of \(D\) in \(\Gamma\) (both with respect to \(S\)). The isoperimetric inequality is then applied to groups of polynomial and exponential growth to derive sharp isoperimetric inequalities in each case, as well as to give bounds and asymptotic estimates of Følner functions.\N\NFor the entire collection see [Zbl 1537.51001].