Algebraic & Geometric Topology

New topological methods to solve equations over groups

Anton Klyachko and Andreas Thom

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We show that the equation associated with a group word w G F2 can be solved over a hyperlinear group G if its content — that is, its augmentation in F2 — does not lie in the second term of the lower central series of F2. 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.

Article information

Algebr. Geom. Topol., Volume 17, Number 1 (2017), 331-353.

Received: 8 December 2015
Revised: 23 March 2016
Accepted: 27 May 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22C05: Compact groups 20F70: Algebraic geometry over groups; equations over groups

equations over groups cohomology of Lie groups


Klyachko, Anton; Thom, Andreas. New topological methods to solve equations over groups. Algebr. Geom. Topol. 17 (2017), no. 1, 331--353. doi:10.2140/agt.2017.17.331.

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