On the submonoid membership problem for HNN extensions of free groups

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.