Infinite words and universal free actions

Abstract. This is the second paper in a series of four, where we take on the unified theory of non-Archimedean group actions, length functions and infinite words. Here, for an arbitrary group G of infinite words over an ordered abelian group Λ we construct a Λ-tree Γ G $\Gamma _G$ equipped with a free action of G. Moreover, we show that Γ G $\Gamma _G$ is a universal tree for G in the sense that it isometrically and equivariantly embeds into every Λ-tree equipped with a free G-action compatible with the original length function on G. Also, for a group G acting freely on a Λ-tree Γ we show how one can easily obtain an embedding of G into the set of reduced infinite words R(Λ,X)$R(\Lambda , X)$ , where the alphabet X is obtained from the action G on Γ.