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.

On small cancellation theory over H.N.N. extensions

1983 ◽  
Vol 94 (1-2) ◽  
pp. 25-47 ◽  
Author(s):  
J. Perraud
Keyword(s):  
Word Problem ◽  
Finite Subset ◽  
Decision Problems ◽  
Normal Closure ◽  
Small Cancellation ◽  
Free Products ◽  
Cancellation Condition ◽  
Small Cancellation Theory ◽  
Amalgamated Free Products ◽  
Small Cancellation Condition

SynopsisSmall cancellation theory has been extended to symmetrized subsets of free products, amalgamated free products and Higman-Neumann-Neumann (H.N.N.) extensions. We though that it was possible to obtain results on decision problems if we could define small cancellation conditions for finite subsets.Sacerdote and Schupp (1974) defined the small cancellation condition C'(l/6) for symmetrized subsets of an H.N.N. extension. We define this condition for finite subsets, with the following properties:For each finite subset X, there is a symmetrized subset X1 with the same normal closure and, if X1 satisfies C'(l/6), then X satisfies C'(l/6).For some H.N.N. extensions, we can decide whether any finite subset satisfies C'(l/6), and, in this case, we can solve the word problem for the corresponding quotient.

scholarly journals GRAPH SMALL CANCELLATION THEORY APPLIED TO ALTERNATING LINK GROUPS

2012 ◽  
Vol 21 (11) ◽  
pp. 1250113
Author(s):  
RÉMI CUNÉO ◽  
HAMISH SHORT
Keyword(s):  
Small Cancellation ◽  
Cancellation Condition ◽  
Small Cancellation Theory ◽  
Link Group ◽  
Finite Sums ◽  
Small Cancellation Condition ◽  
Alternating Links ◽  
Alternating Link

We show that the Wirtinger presentation of a prime alternating link group satisfies a generalized small cancellation condition. This new version of Weinbaum's solution to the word and conjugacy problems for these groups easily extends to finite sums of alternating links.

scholarly journals Generic properties of random subgroups of a free group for general distributions

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Frédérique Bassino ◽  
Cyril Nicaud ◽  
Pascal Weil
Keyword(s):  
Free Group ◽  
Random Variable ◽  
Small Cancellation ◽  
Group Presentation ◽  
Generic Properties ◽  
Cancellation Condition ◽  
Number Of Generators ◽  
International Audience ◽  
General Distributions ◽  
Small Cancellation Condition

International audience We consider a generalization of the uniform word-based distribution for finitely generated subgroups of a free group. In our setting, the number of generators is not fixed, the length of each generator is determined by a random variable with some simple constraints and the distribution of words of a fixed length is specified by a Markov process. We show by probabilistic arguments that under rather relaxed assumptions, the good properties of the uniform word-based distribution are preserved: generically (but maybe not exponentially generically), the tuple we pick is a basis of the subgroup it generates, this subgroup is malnormal and the group presentation defined by this tuple satisfies a small cancellation condition.

scholarly journals Random Subgroups of Acylindrically Hyperbolic Groups and Hyperbolic Embeddings

2017 ◽  
Vol 2019 (13) ◽  
pp. 3941-3980 ◽  
Author(s):  
Joseph Maher ◽  
Alessandro Sisto
Keyword(s):  
Normal Subgroup ◽  
Random Walks ◽  
Free Group ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Finite Collection ◽  
Small Cancellation ◽  
Cancellation Condition ◽  
Asymptotic Probability ◽  
Small Cancellation Condition

Abstract Let $G$ be an acylindrically hyperbolic group. We consider a random subgroup $H$ in $G$, generated by a finite collection of independent random walks. We show that, with asymptotic probability one, such a random subgroup $H$ of $G$ is a free group, and the semidirect product of $H$ acting on $E(G)$ is hyperbolically embedded in $G$, where $E(G)$ is the unique maximal finite normal subgroup of $G$. Furthermore, with control on the lengths of the generators, we show that $H$ satisfies a small cancellation condition with asymptotic probability one.

On T(6)-groups

1987 ◽  
Vol 102 (3) ◽  
pp. 443-451 ◽  
Author(s):  
Mohamed S. El-Mosalamy ◽  
Stephen J. Pride
Keyword(s):  
Small Cancellation ◽  
Cancellation Condition ◽  
Group Presentations ◽  
Definition Of ◽  
Small Cancellation Condition

This paper is concerned with group presentations satisfying the small cancellation condition T(6). The definition of this condition is given in §1·2, together with some examples. Before giving the definition, however, we describe (in §1·1) some material which, to a certain extent, motivated our paper. In § 1·3 we state our main theorem, which provides new solutions to the word and conjugacy problems for T(6)-groups.

scholarly journals Infinitely presented small cancellation groups have the Haagerup property

2015 ◽  
Vol 07 (03) ◽  
pp. 389-406 ◽  
Author(s):  
Goulnara Arzhantseva ◽  
Damian Osajda
Keyword(s):  
Hilbert Space ◽  
Hyperbolic Groups ◽  
Main Step ◽  
Small Cancellation ◽  
Finitely Generated ◽  
Haagerup Property ◽  
Cancellation Condition ◽  
Finitely Generated Groups ◽  
Direct Limits ◽  
Small Cancellation Condition

We prove the Haagerup property (= Gromov's a-T-menability) for finitely generated groups defined by infinite presentations satisfying the C'(1/6)-small cancellation condition. We deduce that these groups are coarsely embeddable into a Hilbert space and that the strong Baum–Connes conjecture holds for them. The result is a first nontrivial advancement in understanding groups with such properties among infinitely presented non-amenable direct limits of hyperbolic groups. The proof uses the structure of a space with walls introduced by Wise. As the main step we show that C'(1/6)-complexes satisfy the linear separation property.

scholarly journals Geometry of Infinitely Presented Small Cancellation Groups and Quasi-homomorphisms

2019 ◽  
Vol 71 (5) ◽  
pp. 997-1018
Author(s):  
Goulnara Arzhantseva ◽  
Cornelia Druţu
Keyword(s):  
Independent Set ◽  
Group Cohomology ◽  
Small Cancellation ◽  
Direct Construction ◽  
Cancellation Condition ◽  
Linearly Independent ◽  
Cross Decomposition ◽  
Small Cancellation Condition ◽  
Free Subgroups

AbstractWe study the geometry of infinitely presented groups satisfying the small cancellation condition $C^{\prime }(1/8)$, and introduce a standard decomposition (called the criss-cross decomposition) for the elements of such groups. Our method yields a direct construction of a linearly independent set of power continuum in the kernel of the comparison map between the bounded and the usual group cohomology in degree 2, without the use of free subgroups and extensions.

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