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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 F2 be the free group generated by x and y. In this article, we prove that the commutator of xm and yn 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 F2 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