  We study the set of $k$-abelian critical exponents of all Sturmian words. It
has been proven that in the case $k = 1$ this set coincides with the Lagrange
spectrum. Thus the sets obtained when $k > 1$ can be viewed as generalized
Lagrange spectra. We characterize these generalized spectra in terms of the
usual Lagrange spectrum and prove that when $k > 1$ the spectrum is a dense
non-closed set. This is in contrast with the case $k = 1$, where the spectrum
is a closed set containing a discrete part and a half-line. We describe
explicitly the least accumulation points of the generalized spectra. Our
geometric approach allows the study of $k$-abelian powers in Sturmian words by
means of continued fractions.
