  This paper introduces Markov chains and processes over nonabelian free groups
and semigroups. We prove a formula for the $f$-invariant of a Markov chain over
a free group in terms of transition matrices that parallels the classical
formula for the entropy a Markov chain. Applications include free group
analogues of the Abramov-Rohlin formula for skew-product actions and
Yuzvinskii's addition formula for algebraic actions.

Dynamical Systems