Infinite words and universal free actions

This is the second paper in a series of four,wherewe 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 . we construct a A-tree I_G equipped with a free action of G. Moreover, we show that .G is a universal tree for G in the sense that it isometrically and equivariantly embeds into every A-tree equipped with a free G-action compatible with the original length function on G. Also, for a group G acting freely on a A-tree . we show how one can easily obtain an embedding of G into the set of reduced infinite words R(., X), where the alphabet X is obtained from the action G → I.