#### 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.

Notes on file formats
Sucharit Sarkar

Stat-Math Unit, Indian Statistical Institute

Bangalore, India

Email: bmat0212@isibang.ac.in

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