Solving a group equation of length nine

TL;DR

This paper investigates the Levin conjecture for a specific group equation of length nine, applying advanced methods to verify its solvability in most cases within torsion free groups.

Contribution

It extends the verification of the Levin conjecture to a length nine group equation using weight test and curvature distribution methods.

Findings

Levin conjecture holds for the length nine equation in most cases

Methods successfully applied to longer group equations

Some exceptional cases remain unresolved

Abstract

The Levin conjecture was proposed by Levin in 1962 which conjectures the solvability of any group equation with coefficients in a torsion free group. The Levin conjecture is recently shown to hold for group equations of length seven by weight test and curvature distribution. These methods are applied on a group equation of length nine to explore the validity of Levin conjecture. It is found that the Levin conjecture has an affirmative answer for this equation modulo some exceptional cases.