New topological methods to solve equations over groups

Abstract

We show that the equation associated with a group word $w\in G\ast F_2$ can be solved over a hyperlinear group $G$ if its content — that is, its augmentation in $F_2$ — does not lie in the second term of the lower central series of $F_2$. Moreover, if $G$ is finite, then a solution can be found in a finite extension of $G$. The method of proof extends techniques developed by Gerstenhaber and Rothaus, and uses computations in $p$–local homotopy theory and cohomology of compact Lie groups.