New results on the prefix membership problem for one-relator groups

TL;DR

This paper advances understanding of the prefix membership problem in one-relator groups by establishing decidability results for certain classes and introducing a new theory of conservative factorizations, with implications for inverse monoids.

Contribution

It provides new decidability results for the prefix membership problem in specific one-relator groups and introduces a novel theory linking this problem to inverse monoids.

Findings

Decidability of prefix membership problem for certain one-relator groups.

Introduction of a new theory of conservative factorizations.

First example of a one-relator group with undecidable prefix membership problem.

Abstract

In this paper we prove several results regarding decidability of the membership problem for certain submonoids in amalgamated free products and HNN extensions of groups. These general results are then applied to solve the prefix membership problem for a number of classes of one-relator groups which are low in the Magnus-Moldavanski\u{\i} hierarchy. Since the prefix membership problem for one-relator groups is intimately related to the word problem for one-relator special inverse monoids in the $E$-unitary case (as discovered in 2001 by Ivanov, Margolis and Meakin), these results yield solutions of the word problem for several new classes of one-relator special inverse monoids. In establishing these results, we introduce a new theory of conservative factorisations of words which provides a link between the prefix membership problem of a one-relator group and the group of units of the corresponding one-relator special inverse monoid. Finally, we exhibit the first example of a one-relator group, defined by a reduced relator word, that has an undecidable prefix membership problem.