Group extensions over infinite words

TL;DR

This paper introduces a new method to construct and analyze group extensions over infinite words, providing tools to study groups with complex behaviors and decidability properties, including embedding HNN extensions.

Contribution

It generalizes previous constructions by defining $E(A,G)$ for arbitrary groups G and embeds the partial monoid of infinite words, offering new structural insights.

Findings

Decidability of the Word Problem linked to Cyclic Membership Problem in G

Embedding of infinite words partial monoid into $E(A,G)$

Structural results on torsion elements and HNN extensions

Abstract

We construct an extension $E(A,G)$ of a given group $G$ by infinite non-Archimedean words over an discretely ordered abelian group like $Z^n$. This yields an effective and uniform method to study various groups that "behave like $G$". We show that the Word Problem for f.g. subgroups in the extension is decidable if and only if and only if the Cyclic Membership Problem in $G$ is decidable. The present paper embeds the partial monoid of infinite words as defined by Myasnikov, Remeslennikov, and Serbin (Contemp. Math., Amer. Math. Soc., 378:37-77, 2005) into $E(A,G)$. Moreover, we define the extension group $E(A,G)$ for arbitrary groups $G$ and not only for free groups as done in previous work. We show some structural results about the group (existence and type of torsion elements, generation by elements of order 2) and we show that some interesting HNN extensions of $G$ embed naturally in the larger group $E(A,G)$.