On the Asphericity of Length-6 Relative Presentations With Torsion-Free Coefficients

Abstract

An 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 H, x : xh 1 xh 2 xh 3 xh 4 xh 5 xh 6 , provided that each coefficient is torsion free.

On solvability of certain equations of arbitrary length over torsion-free groups

“…

Let G be a non-trivial torsion free group and s(t) = g 1 t 1 g 2 t 2 • • • g n t n = 1 (g i ∈ G, i = ±1) be an equation over G containing no blocks of the form t −1 g i t −1 , g i ∈ G. In this paper we show that s(t) = 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 [6].

…”

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

“…Ivanov [13] has exhibited a connection between asphericity of specific topological spaces and the Zero Divisor Conjecture for which Leary [17] gave a short variant of the proof. Along these lines recently, Kim showed asphericity of certain one-relator presentations [14].In this paper we pursue an algorithmic approach to the Zero Divisor Conjecture. To this end, we explain in the following section why it is reasonable to consider the problem over F 2 , the field that contains exactly two elements.…”

On Solvability of Certain Equations of Arbitrary Length Over Torsion-Free Groups

“…The Theorem 1.2 is related to the Kaplansky zero-divisor conjecture [8]. Following a brief discussion on relative presentations and the weight test in Section 2, we prove Theorem 1.1 in Section 3 and Theorem 1.2 in Section 4 in which we also prove slightly more general results when n equals 2 or 3 in the statement of Theorem 1.2.…”