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

2020 ◽  
pp. 1
Author(s):  
Igor Dolinka ◽  
Robert D. Gray
Keyword(s):  
Membership Problem ◽  
One Relator Groups

SOLUTION OF THE MEMBERSHIP PROBLEM FOR CERTAIN RATIONAL SUBSETS OF ONE-RELATOR GROUPS WITH A SMALL CANCELLATION CONDITION

2012 ◽  
Vol 22 (04) ◽  
pp. 1250027 ◽  
Author(s):  
ARYE JUHÁSZ
Keyword(s):  
Natural Number ◽  
Free Group ◽  
Membership Problem ◽  
Small Cancellation ◽  
Cancellation Condition ◽  
Small Cancellation Theory ◽  
Reduced Word ◽  
One Relator Groups ◽  
Condition C ◽  
Small Cancellation Condition

Let n ≥ 2 be a natural number, X = { x1,…, xn} and let F be the free group, freely generated by X. Let R be a cyclically reduced word in F such that its symmetric closure [Formula: see text] in F satisfies the small cancellation condition C′(1/5) & T(4). Let G be the group presented by [Formula: see text]. A Magnus subsemigroup of G is any subsemigroup of G generated by at most 2n - 1 elements of [Formula: see text]. In this paper we solve the Membership Problem for rational subsets of G which are contained in a Magnus subsemigroup of G, provided that [Formula: see text] satisfies certain combinatorial conditions. We use small cancellation theory with word combinatorics.

scholarly journals Undecidability of the word problem for one-relator inverse monoids via right-angled Artin subgroups of one-relator groups

2019 ◽  
Vol 219 (3) ◽  
pp. 987-1008
Author(s):  
Robert D. Gray
Keyword(s):  
Word Problem ◽  
Inverse Monoid ◽  
Artin Group ◽  
Membership Problem ◽  
Finitely Generated ◽  
Inverse Monoids ◽  
New Construction ◽  
One Relator Groups ◽  
The One

Abstract We prove the following results: (1) There is a one-relator inverse monoid $$\mathrm {Inv}\langle A\,|\,w=1 \rangle $$Inv⟨A|w=1⟩ with undecidable word problem; and (2) There are one-relator groups with undecidable submonoid membership problem. The second of these results is proved by showing that for any finite forest the associated right-angled Artin group embeds into a one-relator group. Combining this with a result of Lohrey and Steinberg (J Algebra 320(2):728–755, 2008), we use this to prove that there is a one-relator group containing a fixed finitely generated submonoid in which the membership problem is undecidable. To prove (1) a new construction is introduced which uses the one-relator group and submonoid in which membership is undecidable from (2) to construct a one-relator inverse monoid $$\mathrm {Inv}\langle A\,|\,w=1 \rangle $$Inv⟨A|w=1⟩ with undecidable word problem. Furthermore, this method allows the construction of an E-unitary one-relator inverse monoid of this form with undecidable word problem. The results in this paper answer a problem originally posed by Margolis et al. (in: Semigroups and their applications, Reidel, Dordrecht, pp. 99–110, 1987).

On the isomorphism problem for one-relator groups

1976 ◽  
Vol 27 (1) ◽  
pp. 484-488 ◽  
Author(s):  
Gerhard Rosenberger ◽  
R. N. Kalia
Keyword(s):  
Isomorphism Problem ◽  
One Relator Groups

scholarly journals One-relator groups and algebras related to polyhedral products

2021 ◽  
pp. 1-20
Author(s):  
Jelena Grbić ◽  
George Simmons ◽  
Marina Ilyasova ◽  
Taras Panov
Keyword(s):  
Homology Group ◽  
Homotopy Theory ◽  
Commutator Subgroup ◽  
Homology Groups ◽  
Homological Characterization ◽  
One Relator Groups ◽  
Combinatorial Condition ◽  
The Moment ◽  
Necessary And Sufficient

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$ , we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $\mathcal {R}_K$ , to be a one-relator group; and for the Pontryagin algebra $H_{*}(\Omega \mathcal {Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterization of these properties. For $RC_K'$ , it is given by a condition on the homology group $H_2(\mathcal {R}_K)$ , whereas for $H_{*}(\Omega \mathcal {Z}_K)$ it is stated in terms of the bigrading of the homology groups of $\mathcal {Z}_K$ .

WHEN CHURCH-ROSSER BECOMES CONTEXT FREE

2007 ◽  
Vol 18 (06) ◽  
pp. 1293-1302 ◽  
Author(s):  
MARTIN KUTRIB ◽  
ANDREAS MALCHER
Keyword(s):  
Linear Time ◽  
Membership Problem ◽  
Closure Properties ◽  
Context Free Language ◽  
Arithmetic Hierarchy ◽  
Context Free ◽  
Free Language

We investigate the intersection of Church-Rosser languages and (strongly) context-free languages. The intersection is still a proper superset of the deterministic context-free languages as well as of their reversals, while its membership problem is solvable in linear time. For the problem whether a given Church-Rosser or context-free language belongs to the intersection we show completeness for the second level of the arithmetic hierarchy. The equivalence of Church-Rosser and context-free languages is Π1-complete. It is proved that all considered intersections are pairwise incomparable. Finally, closure properties under several operations are investigated.

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