Solving word problems in group extensions over infinite words

Volker Diekert, Alexei G. Myasnikov

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

2 Scopus citations

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 ∈ 〈v〉?" is decidable for all v ∈ G. Our methods combine combinatorics on words, string rewriting, and group theory.

Original languageEnglish
Title of host publicationDevelopments in Language Theory - 15th International Conference, DLT 2011, Proceedings
Pages192-203
Number of pages12
DOIs
StatePublished - 2011
Event15th International Conference on Developments in Language Theory, DLT 2011 - Milan, Italy
Duration: 19 Jul 201122 Jul 2011

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6795 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference15th International Conference on Developments in Language Theory, DLT 2011
Country/TerritoryItaly
CityMilan
Period19/07/1122/07/11

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