Ordering groups and the Identity Problem
Corentin Bodart, Laura Ciobanu, George Metcalfe
TL;DR
The Identity Problem asks whether the subsemigroup generated by a finite set contains the identity; this work connects IP to left-order extension and ℓ-group word problems, and proves IP and the Subgroup Problem are decidable for every finitely presented nilpotent group via a convex-geometry criterion that reduces the problem to abelian quotients and finite-index checks. It also shows the Fixed-Target Submonoid Membership Problem is undecidable in some nilpotent groups and provides a polynomial-time algorithm for IP in a fixed nilpotent group using Malcev completions and a Lie-algebra framework, with extensions to metabelian groups such as orientation-preserving affine transformations of the rationals. A comprehensive set of techniques—convex geometry, Malcev completions, and orderability theory—links IP to the Word Problem in lattice-ordered groups and to order-extension problems, yielding new decidability and complexity results across nilpotent and metabelian contexts.
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.