Finitely Generated Amenable Groups

Abstract

This chapter is devoted to the growth and amenability of finitely generated groups. The choice of a finite symmetric generating subset for a finitely generated group defines a word metric on the group and a labelled graph, which is called a Cayley graph. The associated growth function counts the number of group elements in a ball of radius n with respect to the word metric. We define a notion of equivalence for such growth functions and observe that the growth functions associated with different finite symmetric generating subsets are in the same equivalence class (Corollary 6.4.5). This equivalence class is called the growth type of the group. The notions of polynomial, subexponential and exponential growth are introduced in Sect. 6.5. In Sect. 6.7 we give an example of a finitely generated metabelian group with exponential growth. We prove that finitely generated nilpotent groups have polynomial growth (Theorem 6.8.1). In Sect. 6.9 we consider the Grigorchuk group. It is shown that it is an infinite periodic (Theorem 6.9.8), residually finite (Corollary 6.9.5) finitely generated group of intermediate growth (Theorem 6.9.17). In the subsequent section, we show that every finitely generated group of subexponential growth is amenable (Theorem 6.11.2). In Sect. 6.12 we prove the Kesten-Day characterization of amenability (Theorem 6.12.9) which asserts that a group with a finite (not necessarily symmetric) generating subset is amenable if and only if 0 is in the ℓ 2-spectrum of the associated Laplacian. Finally, in Sect. 6.13 we consider the notion of quasi-isometry for not necessarily countable groups and we show that amenability is a quasi-isometry invariant (Theorem 6.13.23).