scholarly journals On singular equations over torsion-free groups

2021 ◽  
pp. 1-30
Author(s):  
Martin Edjvet ◽  
James Howie
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Singular Equation ◽  
Torsion Free Group ◽  
Singular Equations ◽  
Torsion Free

We prove a Freiheitssatz for one-relator products of torsion-free groups, where the relator has syllable length at most [Formula: see text]. This result has applications to equations over torsion-free groups: in particular a singular equation of syllable length at most [Formula: see text] over a torsion-free group has a solution in some overgroup.

scholarly journals On a Nonsingular Equation of Length 9 Over Torsion Free Groups

2019 ◽  
Vol 12 (2) ◽  
pp. 590-604
Author(s):  
M. Fazeel Anwar ◽  
Mairaj Bibi ◽  
Muhammad Saeed Akram
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Torsion Free Group ◽  
Torsion Free

In \cite{levin}, Levin conjectured that every equation is solvable over a torsion free group. In this paper we consider a nonsingular equation $g_{1}tg_{2}t g_{3}t g_{4} t g_{5} t g_{6} t^{-1} g_{7} t g_{8}t \\ g_{9}t^{-1} = 1$ of length $9$ and show that it is solvable over torsion free groups modulo some exceptional cases.

ON SMALL LENGTH EQUATIONS OVER TORSION-FREE GROUPS

1994 ◽  
Vol 04 (04) ◽  
pp. 575-589 ◽  
Author(s):  
MATVEĬ I. PRISHCHEPOV
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Small Length ◽  
Torsion Free Group ◽  
The Absolute ◽  
Torsion Free

Let G be a group, <t> the free group generated by t and let r(t)∈G*<t>. The equation r(t)=1 is said to have a solution over G if it has a solution in some group that contains G. There is a conjecture (attributed to F. Levin) that if G is a torsion-free group, then any equation has a solution over G. In this paper we verify this conjecture in the case when the sum of the absolute values of the exponent of t in r(t) is not greater than six.

scholarly journals ON SOLVABILITY OF CERTAIN EQUATIONS OF ARBITRARY LENGTH OVER TORSION-FREE GROUPS

2020 ◽  
pp. 1-9
Author(s):  
MUHAMMAD FAZEEL ANWAR ◽  
MAIRAJ BIBI ◽  
MUHAMMAD SAEED AKRAM
Keyword(s):  
Free Group ◽  
Zero Divisor ◽  
Free Groups ◽  
Arbitrary Length ◽  
Torsion Free Group ◽  
Torsion Free ◽  
Single Relation

Abstract Let G be a nontrivial torsion-free group and $s\left( t \right) = {g_1}{t^{{\varepsilon _1}}}{g_2}{t^{{\varepsilon _2}}} \ldots {g_n}{t^{{\varepsilon _n}}} = 1\left( {{g_i} \in G,{\varepsilon_i} = \pm 1} \right)$ be an equation over G containing no blocks of the form ${t^{- 1}}{g_i}{t^{ - 1}},{g_i} \in G$ . In this paper, we show that $s\left( t \right) = 1$ has a solution over G provided a single relation on coefficients of s(t) holds. We also generalize our results to equations containing higher powers of t. The later equations are also related to Kaplansky zero-divisor conjecture.

scholarly journals Solving equations of length seven over torsion-free groups

2018 ◽  
Vol 21 (1) ◽  
pp. 147-164
Author(s):  
Mairaj Bibi ◽  
Martin Edjvet
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Torsion Free Group ◽  
Solving Equations ◽  
Torsion Free

AbstractPrishchepov [16] proved that all equations of length at most six over torsion-free groups are solvable. A different proof was given by Ivanov and Klyachko in [12]. This supports the conjecture stated by Levin [15] that any equation over a torsion-free group is solvable. Here it is shown that all equations of length seven over torsion-free groups are solvable.

scholarly journals On zero divisors with small support in group rings of torsion-free groups

2013 ◽  
Vol 16 (5) ◽  
Author(s):  
Pascal Schweitzer
Keyword(s):  
Free Group ◽  
Zero Divisor ◽  
Free Groups ◽  
Group Rings ◽  
Torsion Free Group ◽  
Zero Divisors ◽  
Torsion Free ◽  
Small Support

Abstract.Kaplansky's zero divisor conjecture envisions that for a torsion-free group 

scholarly journals On torsion-free hypercentral groups with all subgroups subnormal

1989 ◽  
Vol 31 (2) ◽  
pp. 193-194
Author(s):  
Howard Smith
Keyword(s):  
Solvable Group ◽  
Free Group ◽  
Free Groups ◽  
Torsion Free Group ◽  
Free Solvable Group ◽  
Published Result ◽  
Positive Results ◽  
Torsion Free

There is no example known of a non-nilpotent, torsion-free group which has all of its subgroups subnormal. It was proved in [3] that a torsion-free solvable group with all of its proper subgroups subnormal and nilpotent is itself nilpotent, but that seems to be the only published result in this area which is concerned specifically with torsion-free groups. Possibly the extra hypothesis that the group be hypercentral is sufficient to ensure nilpotency, though this is certainly not the case for groups with torsion, as was shown in [7]. The groups exhibited in that paper were seen to have hypercentral length ω + 1, and we know from [8] that further restricting the hypercentral length can lead to some positive results. Here we shall prove the following theorem.

ON THE ASPHERICITY OF CERTAIN RELATIVE PRESENTATIONS OVER TORSION-FREE GROUPS

2008 ◽  
Vol 18 (06) ◽  
pp. 979-987 ◽  
Author(s):  
SEONG KUN KIM
Keyword(s):  
Free Group ◽  
Infinite Order ◽  
Zero Divisor ◽  
Free Groups ◽  
Group Presentation ◽  
Torsion Free Group ◽  
Proper Power ◽  
Torsion Free

It is noteworthy to find whether a relative presentation with torsion-free coefficients is aspherical or not, since a nonaspherical relative presentation that defines a torsion-free group would provide a potential counterexample to the Kaplansky zero-divisor conjecture, see [7]. In this paper we investigate asphericity of the relative group presentation [Formula: see text] provided that m1, m2, …, mk are nonzero integers, k ≤ 3, each coefficient has infinite order and the relator is not a proper power.

scholarly journals Tesselations ofS2and equations over torsion-free groups

1995 ◽  
Vol 38 (3) ◽  
pp. 485-493 ◽  
Author(s):  
A. Clifford ◽  
R. Z. Goldstein
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Torsion Free Group ◽  
Torsion Free

LetGbe a torsion free group,Fthe free group generated byt. The equationr(t) = 1 is said to have a solution overGif there is a solution in some group that containsG. In this paper we generalize a result due to Klyachko who established the solution when the exponent sum oftis one.

scholarly journals A just-nonsolvable torsion-free group defined on the binary tree

1999 ◽  
Vol 211 (1) ◽  
pp. 99-114 ◽  
Author(s):  
A.M. Brunner ◽  
Said Sidki ◽  
Ana Cristina Vieira
Keyword(s):  
Free Group ◽  
Binary Tree ◽  
Torsion Free Group ◽  
Torsion Free

scholarly journals Abelian groups with small cotorsion images

1991 ◽  
Vol 50 (2) ◽  
pp. 243-247 ◽  
Author(s):  
Rüdiger Göbel
Keyword(s):  
Homomorphic Image ◽  
Free Group ◽  
Abelian Groups ◽  
Torsion Free Group ◽  
Compact Abelian Groups ◽  
Torsion Free

AbstractEpimorphic images of compact (algebraically compact) abelian groups are called cotorsion groups after Harrison. In a recent paper, Ph. Schultz raised the question whether “cotorsion” is a property which can be recognized by its small cotorsion epimorphic images: If G is a torsion-free group such that every torsion-free reduced homomorphic image of cardinality is cotorsion, is G necessarily cortorsion? In this note we will give some counterexamples to this problem. In fact, there is no cardinal k which is large enough to test cotorsion.

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