Algebraic and Geometric Topology 4 (2004), paper no. 27, pages 595-602.

Commutators and squares in free groups

Sucharit Sarkar

Abstract. 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 mn 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.

Keywords. Commutators, free groups, products of commutators

AMS subject classification. Primary: 20F12. Secondary: 57M07.

DOI: 10.2140/agt.2004.4.595

E-print: arXiv:math.GR/0409087

Submitted: 5 November 2003. (Revised: 25 March 2004.) Accepted: 12 July 2004. Published: 16 August 2004.

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Sucharit Sarkar
Stat-Math Unit, Indian Statistical Institute
Bangalore, India

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