  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.
