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

2019/03/15 by M. Fazeel Anwar, Anwar, M. Fazeel, Mairaj Bibi +3 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1903.06503

openalex publication_date 2019/03/15 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28

Abstract

Let G be a non-trivial torsion free group and\ns(t)=g₁t^\ε₁g₂t^\ε₂ \⋯ g_nt^\ε_n=1\n ; (g_i \∈ G, \ε_i=\± 1) be an equation over G containing no\nblocks of the form t^-1g_it^-1, ; g_i \∈ G. In this paper we show\nthat s(t)=1 has a solution over G provided a single relation on\ncoefficients of s(t) holds. We also generalize our results to equations\ncontaining higher powers of t. The later equations are also related to\nKaplansky zero-divisor conjecture citeK.\n

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• Unimodular equations which do not preserve the derived length of a group

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