Group extensions over infinite words
Volker Diekert, Alexei Myasnikov

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 of a given group by infinite non-Archimedean words over an discretely ordered abelian group like . This yields an effective and uniform method to study various groups that "behave like ". 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 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 . Moreover, we define the extension group for arbitrary groups 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 embed naturally…
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