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

Nonabelian free group actions: Markov processes, the Abramov-Rohlin formula and Yuzvinskii's formula