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

Dolinka, Igor and Gray, Robert (2021) New results on the prefix membership problem for one-relator groups. Transactions of the American Mathematical Society, 374 (6). pp. 4309-4358. ISSN 0002-9947

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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.

Item Type: Article
Uncontrolled Keywords: one-relator group,prefix membership problem,special inverse monoid,word problem,mathematics(all),applied mathematics ,/dk/atira/pure/subjectarea/asjc/2600
Faculty \ School: Faculty of Science > School of Mathematics
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Depositing User: LivePure Connector
Date Deposited: 10 Nov 2020 01:17
Last Modified: 22 Oct 2022 07:25
DOI: 10.1090/tran/8338

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