The power graph of a torsion-free group determines the directed power graph

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.

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Abelian groups with small cotorsion images

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700032729 ◽

1991 ◽

Vol 50 (2) ◽

pp. 243-247 ◽

Cited By ~ 22

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

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091505001367 ◽

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.

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Periodic cohomology and subgroups with bounded Bredon cohomological dimension

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004107000837 ◽

2008 ◽

Vol 144 (2) ◽

pp. 329-336 ◽

Cited By ~ 2

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.

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