Equations of length five over groups
This work considers the problem of Equations Over Groups and settles the KL- conjecture for equations of length five. Firstly, the problem of equations over groups is stated and discussed and the results, which were up to now obtained, are presented. Then, by way of contradiction, it is assumed that for the remaining cases of equations of length five a solution does not exist. The methodology adopted uses the combinatorial and topological arguments of relative diagrams. If D is a relative diagram representing the counter example, all types of interior regions of positive curvature are listed for each type of equation of length five. For each interior region of positive curvature, one region of negative curvature is found and the positive curvature is added to it to obtain the total curvature in the interior of diagram D. In the final chapter the curvature of the interior of D is added to the curvature of the boundary regions to obtain the total curvature of the diagram. It is proved that the total curvature of 4pi cannot be achieved, our desired contradiction, and therefore equations of length five have a solution.