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

TL;DR

This paper develops a framework for Markov processes over nonabelian free groups, deriving formulas for their invariants and extending classical entropy and algebraic action results to this noncommutative setting.

Contribution

It introduces Markov chains over free groups and establishes formulas for their invariants, extending classical entropy and algebraic action formulas to nonabelian free group actions.

Findings

Derived a formula for the $f$-invariant of Markov chains over free groups.

Extended Abramov-Rohlin formula to free group skew-product actions.

Established Yuzvinskii's addition formula for algebraic actions in the free group context.

Abstract

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.