ON SOLVABILITY OF CERTAIN EQUATIONS OF ARBITRARY LENGTH OVER TORSION-FREE GROUPS
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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.
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2008 ◽
Vol 18
(06)
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pp. 979-987
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2019 ◽
Vol 12
(2)
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pp. 590-604
2008 ◽
Vol 51
(1)
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pp. 201-214
1994 ◽
Vol 04
(04)
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pp. 575-589
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1995 ◽
Vol 38
(3)
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pp. 485-493
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