KMS Spectra for Group Actions on Compact Spaces

Abstract

Given a topologically free action of a countable group G on a compact metric space X, there is a canonical correspondence between continuous 1-cocycles for this group action and diagonal 1-parameter groups of automorphisms of the reduced crossed product C$^*$-algebra. The KMS spectrum is defined as the set of inverse temperatures for which there exists a KMS state. We prove that the possible KMS spectra depend heavily on the nature of the acting group G. For groups of subexponential growth, we prove that the only possible KMS spectra are $\{0\}$, $[0,+\infty )$, $(-\infty ,0]$ and ${\mathbb {R}}$. For certain wreath product groups, which are amenable and of exponential growth, we prove that any closed subset of ${\mathbb {R}}$ containing zero arises as KMS spectrum. Finally, for certain nonamenable groups including the free group with infinitely many generators, we prove that any closed subset may arise. Besides uncovering a surprising relation between geometric group theoretic properties and KMS spectra, our results provide two simple C$^*$-algebras with the following universality property: any closed subset (containing, resp. not containing zero) arises as the KMS spectrum of a 1-parameter group of automorphisms of this C$^*$-algebra.