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

Pages (fromto)  43094358 
Number of pages  50 
Journal  Transactions of the American Mathematical Society 
Volume  374 
Issue number  6 
Early online date  30 Mar 2021 
DOIs  
Publication status  Published  2021 
Keywords
 Onerelator group
 Prefix membership problem
 Special inverse monoid
 Word problem
Profiles

Robert Gray
 School of Mathematics  Reader in Pure Mathematics
 Algebra and Combinatorics  Member
 Logic  Member
Person: Research Group Member, Academic, Teaching & Research
Projects
 1 Finished