Levin Conjecture for Group Equations of Length 9

Muhammad Saeed Akram, Maira Amjid, Sohail Iqbal

Abstract

Levin conjecture states that every group equation is solvable over any torsion free group. The conjecture is shown to hold true for group equation of length seven using weight test and curvature distribution method. Recently, these methods are used to show that Levin conjecture is true for some group equations of length eight and nine modulo some exceptional cases. In this paper, we show that Levin conjecture holds true for a group equation of length nine modulo 2 exceptional cases. In addition, we present the list of cases that are still open for two more equations of length nine.

Keywords

Group equations; torsion-free groups; relative group presentations; asphericity; weight test; curvature distribution

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