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

In this paper we prove several results regarding decidability ofthe membership problem for certain submonoids in amalgamated free prod-ucts and HNN extensions of groups. These general results are then appliedto solve the prefix membership problem for a number of classes of one-relatorgroups which are low in the Magnus–Moldavanski ̆ı hierarchy. Since the pre-fix membership problem for one-relator groups is intimately related to theword problem for one-relator special inverse monoids in theE-unitary case(as discovered in 2001 by Ivanov, Margolis and Meakin), these results yieldsolutions of the word problem for several new classes of one-relator specialinverse monoids. In establishing these results, we introduce a new theory ofconservative factorisations of words which provides a link between the pre-fix membership problem of a one-relator group and the group of units of thecorresponding one-relator special inverse monoid. Finally, we exhibit the firstexample of a one-relator group, defined by a reduced relator word, that has anundecidable prefix membership problem.