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We study a notion of entropy for probability measure preserving actions offinitely generated free groups, called f-invariant entropy, introduced by LewisBowen. In the degenerate case, the f-invariant entropy is negative infinity. Inthis paper we investigate the qualitative consequences of having finitef-invariant entropy. We find three main properties of such actions. First, thestabilizers occurring in factors of such actions are highly restricted.Specifically, the stabilizer of almost every point must be either trivial or offinite index. Second, such actions are very chaotic in the sense that, when thespace is not essentially countable, every non-identity group element acts withinfinite Kolmogorov--Sinai entropy. Finally, we show that such actions displaybehavior reminiscent of the Howe--Moore property. Specifically, if the actionis ergodic then there is an integer n such that for every non-trivial normalsubgroup K the number of K-ergodic components is at most n. Our results arebased on a new formula for f-invariant entropy.

Finite entropy actions of free groups, rigidity of stabilizers, and a Howe-Moore type phenomenon