New topological methods to solve equations over groups

Klyachko, Anton; Thom, Andreas

doi:10.2140/agt.2017.17.331

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New topological methods to solve equations over groups

Anton Klyachko and Andreas Thom

Algebraic & Geometric Topology 17 (2017) 331–353

DOI: 10.2140/agt.2017.17.331

Abstract

We show that the equation associated with a group word $w \in G * F_{2}$ 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

References

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

Volume 17, issue 1 (2017)