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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].

##### MSC:
 43A07 Means on groups, semigroups, etc.; amenable groups
##### Keywords:
amenability; Følner property; random walks