The Identity Problem in nilpotent groups of bounded class

TL;DR

This paper proves that the Identity and Group Problems are decidable in polynomial time for finitely generated subsemigroups of certain nilpotent matrix groups, extending previous results to higher class groups.

Contribution

It extends decidability results for the Identity and Group Problems to nilpotent groups of class up to ten, including non-commutative matrix groups, and provides a condition for generalization to higher classes.

Findings

Decidability of the Identity and Group Problems in polynomial time for class ≤ 10.

Extension of previous work from commutative groups and SL(2, Z) to broader nilpotent groups.

A verifiable condition for extending results to groups of class greater than 10.

Abstract

Let $G$ be a unitriangular matrix group of nilpotency class at most ten. We show that the Identity Problem (does a semigroup contain the identity matrix?) and the Group Problem (is a semigroup a group?) are decidable in polynomial time for finitely generated subsemigroups of $G$. Our decidability results also hold when $G$ is an arbitrary finitely generated nilpotent group of class at most ten. This extends earlier work of Babai et al. on commutative matrix groups (SODA'96) and work of Bell et al. on $\mathsf{SL}(2, \mathbb{Z})$ (SODA'17). Furthermore, we formulate a sufficient condition for the generalization of our results to nilpotent groups of class $d > 10$. For every such $d$, we exhibit an effective procedure that verifies this condition in case it is true.