New topological methods to solve equations over groups

Abstract

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.

17 pages, no figures; v2 contains corrections according to referee report

References in corpus (2)

• The number of non-solutions to an equation in a group and non-topologizable torsion-free groups
• On the lattice of normal subgroups in ultraproducts of compact simple groups

Cited by in corpus (6)

• Yet another Freiheitssatz: Mating finite groups with locally indicable ones
• Equations over solvable groups
• On $p$-nonsingular systems of equations over solvable groups
• Word images in symmetric and classical groups of Lie type are dense
• Infinite systems of equations in abelian and nilpotent groups
• Mixed identities for oligomorphic automorphism groups