  The abelian critical exponent of an infinite word $w$ is defined as the
maximum ratio between the exponent and the period of an abelian power occurring
in $w$. It was shown by Fici et al. that the set of finite abelian critical
exponents of Sturmian words coincides with the Lagrange spectrum. This spectrum
contains every large enough positive real number. We construct words whose
abelian critical exponents fill the remaining gaps, that is, we prove that for
each nonnegative real number $\theta$ there exists an infinite word having
abelian critical exponent $\theta$. We also extend this result to the
$k$-abelian setting.
