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arXiv · 2411.15639

Ordering groups and the Identity Problem

Corentin Bodart·Laura Ciobanu·George Metcalfe
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Abstract

In this paper, the Identity Problem for certain groups, which asks if the subsemigroup generated by a given finite set of elements contains the identity element, is related to problems regarding ordered groups. Notably, the Identity Problem for a torsion-free nilpotent group corresponds to the problem asking if a given finite set of elements extends to the positive cone of a left-order on the group, and thereby also to the Word Problem for a related lattice-ordered group. A new (independent) proof is given showing that the Identity and Subgroup Problems are decidable for every finitely presented nilpotent group, establishing also the decidability of the Word Problem for a family of lattice-ordered groups. A related problem, the Fixed-Target Submonoid Membership Problem, is shown to be undecidable in nilpotent groups. Decidability of the Normal Identity Problem (with `subsemigroup' replaced by `normal subsemigroup') for free nilpotent groups is established using the (known) decidability of the Word Problem for certain lattice-ordered groups. Connections between orderability and the Identity Problem for a class of torsion-free metabelian groups are also explored.

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BibTeXRIS

Corentin Bodart, Laura Ciobanu, George Metcalfe. 2024-11-23. Ordering groups and the Identity Problem. https://doi.org/10.1016/j.jalgebra.2025.10.051

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