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 are then appliedto solve the prefix a number classes one-relatorgroups which low Magnus–Moldavanski i hierarchy. Since pre-fix one-relator groups is intimately related to theword special inverse monoids theE-unitary case(as discovered 2001 by Ivanov, Margolis Meakin), these yieldsolutions word new specialinverse monoids. establishing results, introduce theory ofconservative factorisations words provides link between group units thecorresponding monoid. Finally, exhibit firstexample group, defined reduced relator word, that has anundecidable problem.