Effective equation solving, constraints and growth in virtually abelian groups

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Ciobanu, Laura, Evetts, Alex and Levine, Alex (2025) Effective equation solving, constraints and growth in virtually abelian groups. SIAM Journal on Applied Algebra and Geometry, 9 (1). pp. 235-260. ISSN 2470-6566

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Abstract

In this paper, we study the satisfiability and solutions of group equations when combinatorial, algebraic, and language-theoretic constraints are imposed on the solutions. We show that the solutions to equations with length, lexicographic order, abelianization, or context-free constraints added can be effectively produced in finitely generated virtually abelian groups. Crucially, we translate each of the constraints above into a rational set in an effective way, and so reduce each problem to solving equations with rational constraints, which is decidable and well understood in virtually abelian groups. A byproduct of our results is that the growth series of a virtually abelian group, with respect to any generating set and any weight, is effectively computable. This series is known to be rational by the work of Benson [Invent. Math., 73 (1983), pp. 251–269], but his approach is not constructive.

Item Type: Article
Uncontrolled Keywords: context-free language,equations in groups,growth of groups,rational set,semilinear set,virtually abelian groups,applied mathematics,geometry and topology,algebra and number theory ,/dk/atira/pure/subjectarea/asjc/2600/2604
Faculty \ School: Faculty of Science > School of Engineering, Mathematics and Physics
UEA Research Groups: Faculty of Science > Research Groups > Algebra, Number Theory, Logic, and Representations (ANTLR)
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Depositing User: LivePure Connector
Date Deposited: 10 Jan 2025 01:00
Last Modified: 28 Mar 2025 13:08
URI: https://ueaeprints.uea.ac.uk/id/eprint/98132
DOI: 10.1137/23M1604679

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