Geodesic
ON SOLVABILITY OF CERTAIN EQUATIONS OF ARBITRARY LENGTH OVER TORSION-FREE GROUPS
Glasgow mathematical journal, Tome 63 (2021) no. 3, pp. 651-659

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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. 
ANWAR, MUHAMMAD FAZEEL; BIBI, MAIRAJ; AKRAM, MUHAMMAD SAEED. ON SOLVABILITY OF CERTAIN EQUATIONS OF ARBITRARY LENGTH OVER TORSION-FREE GROUPS. Glasgow mathematical journal, Tome 63 (2021) no. 3, pp. 651-659. doi: 10.1017/S0017089520000427
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     title = {ON {SOLVABILITY} {OF} {CERTAIN} {EQUATIONS} {OF} {ARBITRARY} {LENGTH} {OVER} {TORSION-FREE} {GROUPS}},
     journal = {Glasgow mathematical journal},
     pages = {651--659},
     year = {2021},
     volume = {63},
     number = {3},
     doi = {10.1017/S0017089520000427},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089520000427/}
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