New topological methods to solve equations over groups

TL;DR

This paper introduces new topological techniques to solve equations over groups, demonstrating solutions exist under specific conditions and extending previous methods with advanced homotopy and cohomology tools.

Contribution

It develops novel topological methods for solving group equations, especially over hyperlinear groups, using $p$-local homotopy theory and cohomology of Lie groups.

Findings

Solutions exist for certain group equations over hyperlinear groups.

Finite solutions can be found in finite extensions when the group is finite.

The methods extend previous techniques by incorporating advanced topological computations.

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.