Commutators and squares in free groups

Let $F_{2}$ be the free group generated by $x$ and $y$ . In this article, we prove that the commutator of $x^{m}$ and $y^{n}$ is a product of two squares if and only if $m n$ is even. We also show using topological methods that there are infinitely many obstructions for an element in $F_{2}$ to be a product of two squares.

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Keywords

Commutators, free groups, products of commutators

Mathematical Subject Classification 2000

Primary: 20F12

Secondary: 57M07

References

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Publication

Received: 2 January 2003

Revised: 25 March 2004

Accepted: 12 July 2004

Published: 5 November 2004

Authors

Sucharit Sarkar
Stat-Math Unit Indian Statistical Institute Bangalore India

Volume 4, issue 1 (2004)

Sucharit Sarkar

Abstract