  We show that the equation associated with a group word $w \in G \ast {\mathbf
F}_2$ can be solved over a hyperlinear group $G$ if its content - that is its
augmentation in ${\mathbf F}_2$ - does not lie in the second term of the lower
central series of ${\mathbf F}_2$. 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.
