Commutators and squares in free groups

Sarkar, Sucharit

doi:10.2140/agt.2004.4.595

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Commutators and squares in free groups

Sucharit Sarkar

Algebraic & Geometric Topology 4 (2004) 595–602

DOI: 10.2140/agt.2004.4.595

arXiv: math.GR/0409087

Abstract

Let $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi mathvariant="double-struck">F</mi></mrow><mrow><mn>2</mn></mrow></msub></math>$ be the free group generated by $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>y</mi></math>$ . In this article, we prove that the commutator of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>x</mi></mrow><mrow><mi>m</mi></mrow></msup></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msup></math>$ is a product of two squares if and only if $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>m</mi><mi>n</mi></math>$ is even. We also show using topological methods that there are infinitely many obstructions for an element in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi mathvariant="double-struck">F</mi></mrow><mrow><mn>2</mn></mrow></msub></math>$ to be a product of two squares.

Keywords

Commutators, free groups, products of commutators

Mathematical Subject Classification 2000

Primary: 20F12

Secondary: 57M07

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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)