Ordering groups and the Identity Problem

The Identity Problem — deciding if the subsemigroup generated by a given finite set of elements of a group contains the identity element — is shown in this paper to correspond, for certain classes, to decision problems about ordering groups. Notably, the Identity Problem for a torsion-free nilpotent group corresponds both to the problem of deciding if a given finite set of elements extends to the positive cone of a left-order on the group, and to the Word Problem for a related lattice-ordered group.

A new (independent) proof is given of the decidability of the Identity and Subgroup Problems for every finitely presented nilpotent group (initially proved by Shafrir in 2024), establishing also the decidability of the Word Problem for a family of lattice-ordered groups. In contrast, it is shown that the related Fixed-Target Submonoid Membership Problem is 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.