scholarly journals ON THE ASPHERICITY OF LENGTH-6 RELATIVE PRESENTATIONS WITH TORSION-FREE COEFFICIENTS

2008 ◽  
Vol 51 (1) ◽  
pp. 201-214
Author(s):  
Seong Kun Kim
Keyword(s):  
Free Group ◽  
Zero Divisor ◽  
Point Of View ◽  
Torsion Free Group ◽  
Torsion Free

AbstractAn interesting result of Ivanov implies that a non-aspherical relative presentation that defines a torsion-free group would provide a potential counterexample to the Kaplansky zero-divisor conjecture. In this point of view, we prove the asphericity of the length-6 relative presentation $\langle H,x: xh_1xh_2xh_3xh_4xh_5xh_6\rangle$, provided that each coefficient is torsion free.

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 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 

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 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 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.

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.

Periodic cohomology and subgroups with bounded Bredon cohomological dimension

2008 ◽  
Vol 144 (2) ◽  
pp. 329-336 ◽  
Author(s):  
JANG HYUN JO ◽  
BRITA E. A. NUCINKIS
Keyword(s):  
Free Group ◽  
Amenable Group ◽  
Cohomological Dimension ◽  
Dimensional Model ◽  
Torsion Free Group ◽  
Bredon Cohomology ◽  
Finite Dimensional ◽  
Torsion Free

AbstractMislin and Talelli showed that a torsion-free group in$\HF$with periodic cohomology after some steps has finite cohomological dimension. In this note we look at similar questions for groups with torsion by considering Bredon cohomology. In particular we show that every elementary amenable group acting freely and properly on some$\R^n$×Smadmits a finite dimensional model for$\E$G.

scholarly journals Torsion-free group without unique product property

1987 ◽  
Vol 108 (1) ◽  
pp. 116-126 ◽  
Author(s):  
Eliyahu Rips ◽  
Yoav Segev
Keyword(s):  
Free Group ◽  
Torsion Free Group ◽  
Product Property ◽  
Unique Product ◽  
Torsion Free
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