Finite phase transitions in countable abelian groups

Abstract

Let A be an infinite set that generates a group G. The sphere SA(r) is the set of elements of G for which the word length with respect to A is exactly r. We say G admits all finite transitions if for every r ≥ 2 and every finite symmetric subset \({W \subset G{\setminus}\{e\}}\), there exists an A with SA(r) = W. In this paper we determine which countable abelian groups admit all finite transitions. We also show that \({\mathbb{R}^n}\) and the finitary symmetric group on \({\mathbb{N}}\) admit all finite transitions.