Ordering groups and the Identity Problem

TL;DR

This paper explores the decidability of the Identity Problem in various groups, linking it to orderability and the Word Problem, and provides new proofs and results for nilpotent and metabelian groups.

Contribution

It introduces a new proof for decidability of the Identity and Subgroup Problems in finitely presented nilpotent groups and explores their connections to ordered groups and the Word Problem.

Findings

Decidability of the Identity and Subgroup Problems for finitely presented nilpotent groups.

Undecidability of the Fixed-Target Submonoid Membership Problem in nilpotent groups.

Decidability of the Normal Identity Problem for free nilpotent groups.

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.