Solving Word Problems in Group Extensions over Infinite Words

Abstract

Non-Archimedean words have been introduced as a new type of infinite words which can be investigated through classical methods in combinatorics on words due to a length function. The length function, however, takes values in the additive group of polynomials ℤ[t] (and not, as traditionally, in ℕ), which yields various new properties. Non-Archimedean words allow to solve a number of algorithmic problems in geometric and algorithmic group theory. There is a connection to the first-order theory in free groups (Tarski Problems), too.

In the present paper we provide a general method to use infinite words over a discretely ordered abelian group as a tool to investigate certain group extensions for an arbitrary group G. The central object is a group E(A,G) which is defined in terms of a non-terminating, but confluent rewriting system. The group G as well as some natural HNN-extensions of G embed into E(A,G) (and still ”behave like” G), which makes it interesting to study its algorithmic properties. The main result characterizes when the Word Problem (WP ) is decidable in all finitely generated subgroups of E(A,G). We show that this property holds if and only if the Cyclic Membership Problem ”$u \in \left< \mathinner{v} \right>$?” is decidable for all v ∈ G. Our methods combine combinatorics on words, string rewriting, and group theory.