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Previous work introduced two measure-conjugacy invariants: the $f$-invariant(for actions of free groups) and $\Sigma$-entropy (for actions of soficgroups). The purpose of this paper is to show that the $f$-invariant is aspecial case of $\Sigma$-entropy. There are two applications: the $f$-invariantis invariant under group automorphisms and there is a uniform lower bound onthe $f$-invariant of a factor in terms of the original system.

The ergodic theory of free group actions: entropy and the $f$-invariant