Groups with free regular length functions in Z ⁿ

This is the first paper in a series of three where we take on the unified theory of non-Archimedean group actions, length functions and infinite words. Our main goal is to show that group actions on Z ⁿ-trees give one a powerful tool to study groups. All finitely generated groups acting freely on R-trees also act freely on some Z ⁿ-trees, but the latter ones form a much larger class. The natural effectiveness of all constructions for Z ⁿ-actions (which is not the case for R-trees) comes along with a robust algorithmic theory. In this paper we describe the algebraic structure of finitely generated groups acting freely and regularly on Z ⁿ-trees and give necessary and sufficient conditions for such actions.