Infinite words and universal free actions

TL;DR

This paper develops a universal $ extLambda$-tree for groups acting freely via infinite words, unifying non-Archimedean group actions, length functions, and tree structures in a comprehensive framework.

Contribution

It constructs a universal $ extLambda$-tree for any group of infinite words with a free action, extending the theory of non-Archimedean group actions.

Findings

Constructed a $ extLambda$-tree with a free $G$-action for arbitrary groups of infinite words.

Proved the universality of the constructed $ extLambda$-tree in embedding all compatible free actions.

Unified the theory of non-Archimedean group actions, length functions, and infinite words.

Abstract

This is the second paper in a series of three, 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 $\Lambda$ we construct a $\Lambda$-tree $\Gamma_G$ equipped with a free action of $G$. Moreover, we show that $\Gamma_G$ is a universal tree for $G$ in the sense that it isometrically embeds in every $\Lambda$-tree equipped with a free $G$-action compatible with the original length function on $G$.