Volume 4, issue 1 (2004)

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Commutators and squares in free groups

Sucharit Sarkar

Algebraic & Geometric Topology 4 (2004) 595–602
 arXiv: math.GR/0409087
Abstract

Let ${\mathbb{F}}_{2}$ 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 ${\mathbb{F}}_{2}$ to be a product of two squares.

Keywords
Commutators, free groups, products of commutators
Primary: 20F12
Secondary: 57M07