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.
Categories: math.GR math.AT