2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/226844This 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.39 pages. The introduction has been rewritten and minor errors correctedDynamical Systems37A35text