#### Volume 17, issue 1 (2017)

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New topological methods to solve equations over groups

### Anton Klyachko and Andreas Thom

Algebraic & Geometric Topology 17 (2017) 331–353
##### 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.

##### Keywords
equations over groups, cohomology of Lie groups
##### Mathematical Subject Classification 2010
Primary: 22C05, 20F70
##### Publication
Received: 8 December 2015
Revised: 23 March 2016
Accepted: 27 May 2016
Published: 26 January 2017
##### Authors
 Anton Klyachko Faculty of Mechanics and Mathematics Moscow State University Leninskie Gory Moscow 119991 Russia Andreas Thom Institut für Geometrie TU Dresden D-01062 Dresden Germany