Infinite words and universal free actions
Olga Kharlampovich, Alexei Myasnikov, Denis Serbin

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 of infinite words over an ordered abelian group we construct a -tree equipped with a free action of . Moreover, we show that is a universal tree for in the sense that it isometrically embeds in every -tree equipped with a free -action compatible with the original length function on .
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