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

Part of: Special aspects of infinite or finite groups Low-dimensional topology

Published online by Cambridge University Press:  13 October 2020

MUHAMMAD FAZEEL ANWAR ,
MAIRAJ BIBI  and
MUHAMMAD SAEED AKRAM
MUHAMMAD FAZEEL ANWAR
Affiliation:
Department of Mathematics, Sukkur IBA University, e-mail:fazeel.anwar@iba-suk.edu.pk
MAIRAJ BIBI
Affiliation:
Department of Mathematics, COMSATS University, Islamabad, Pakistan, e-mail:mairaj_maths@comsats.edu.pk
MUHAMMAD SAEED AKRAM
Affiliation:
Department of Mathematics, Khawaja Fareed UEIT, e-mail:mrsaeedakram@gmail.com
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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.

MSC classification

Primary: 20F05: Generators, relations, and presentations
Secondary: 57M07: Topological methods in group theory
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 3 , September 2021 , pp. 651 - 659
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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