Amenable groups, isoperimetric profiles and random walks. (English) Zbl 0934.43001

Cossey, John (ed.) et al., Geometric group theory down under. Proceedings of a special year in geometric group theory, Canberra, Australia, July 14-19, 1996. Berlin: de Gruyter. 293-316 (1999).
This important article is a comprehensive contribution to control the hierarchy of several remarkable classes of discrete groups. Let \(\Gamma\) be a (finitely generated) group with a finite symmetric generating subset \(S\). For \(\Omega \subset \Gamma\), a boundary is defined by \[ \partial \Omega= \{\gamma\in \Gamma : (\exists s \in S) \gamma s \in \Gamma \setminus \Omega \}. \] Følner’s property says that \(\Gamma\) is amenable if and only if there exists a sequence \((\Omega _n)\) of finite subsets in \(\Gamma\) such that \[ \lim _{n \rightarrow \infty} \frac{|\partial \Omega _n|}{|\Omega _n|} = 0. \] Let now \[ I(n) = \max _{m \leq n} \min _{|\Omega |= m} |\partial \Omega|; \] \(\Gamma\) is nonamenable if and only if \(I(n) \sim n\). Whereas all finitely generated groups with polynomial growth are amenable, the authors make the interesting observation that there exist amenable groups for which \(I(n)\) is not bounded by \(\frac{n}{\text{Log }n}.\) If \(x, y \in \Gamma\), define \(K(x,y) = \frac{1}{|S|}\) if there exists \( s \in S \) such that \(xs = y \) and \(K(x,y) = 0\) otherwise; also suppose that \(K(x,y) = K(y,x)\) and \(\sum _{s \in S} K(e,s) = 1\). Let \[ p_t(x,y) = \sum _{z_0, z_1, \dots z_t} K(z_0, z_1) \dots K(z_{t-1}, z_t) \] with the conditions \(z_0 = x, z_t = y\). This definition admits an obvious interpretation in the language of random walks. The hierarchy ruled by a partial ordering \[ H \leq \Gamma \leftrightarrow (p_t)_H \geq (p_t)_{\Gamma} \] is studied. The authors also formulate a collection of conjectures and speculations.
For the entire collection see [Zbl 0910.00040].


43A07 Means on groups, semigroups, etc.; amenable groups