On the submonoid membership problem for HNN extensions of free groups

TL;DR

This paper investigates the submonoid membership problem in HNN extensions of free groups and applies these findings to solve the word problem in specific one-relator inverse monoids, introducing new decidability techniques.

Contribution

It introduces new decidability conditions for submonoid membership in HNN extensions and applies them to solve the word problem in certain one-relator inverse monoids.

Findings

Decidable membership conditions for submonoids in HNN extensions.

Solution to the word problem in a specific example of one-relator inverse monoids.

Extension of previous methods enabling resolution of previously unsolvable cases.

Abstract

We study membership problems in HNN extensions of free groups and then apply these results to solve the word problem in certain families of one-relator inverse monoids. In more detail, we consider HNN extensions where the defining isomorphism produces a bijection between subsets of a basis of the free group. Within such HNN extensions we identify natural conditions on submonoids of this group that suffice for membership in that submonoid to be decidable. We show that these results can then be applied to solve the prefix membership problem in certain one-relator groups which via results of Ivanov, Margolis and Meakin $(2001)$ then give solutions to the word problem for the corresponding one-relator inverse monoid. In particular our new techniques allow us to solve the word problem in an example (Example $7.6$) from Dolinka and Gray $(2021)$ which previous methods had not been able to resolve.