  We study a notion of entropy for probability measure preserving actions of
finitely generated free groups, called f-invariant entropy, introduced by Lewis
Bowen. In the degenerate case, the f-invariant entropy is negative infinity. In
this paper we investigate the qualitative consequences of having finite
f-invariant entropy. We find three main properties of such actions. First, the
stabilizers occurring in factors of such actions are highly restricted.
Specifically, the stabilizer of almost every point must be either trivial or of
finite index. Second, such actions are very chaotic in the sense that, when the
space is not essentially countable, every non-identity group element acts with
infinite Kolmogorov--Sinai entropy. Finally, we show that such actions display
behavior reminiscent of the Howe--Moore property. Specifically, if the action
is ergodic then there is an integer n such that for every non-trivial normal
subgroup K the number of K-ergodic components is at most n. Our results are
based on a new formula for f-invariant entropy.
