## Algebraic & Geometric Topology

- Algebr. Geom. Topol.
- Volume 4, Number 1 (2004), 595-602.

### Commutators and squares in free groups

#### 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.

#### Article information

**Source**

Algebr. Geom. Topol., Volume 4, Number 1 (2004), 595-602.

**Dates**

Received: 2 January 2003

Revised: 25 March 2004

Accepted: 12 July 2004

First available in Project Euclid: 21 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.agt/1513882489

**Digital Object Identifier**

doi:10.2140/agt.2004.4.595

**Mathematical Reviews number (MathSciNet)**

MR2077678

**Zentralblatt MATH identifier**

1055.20025

**Subjects**

Primary: 20F12: Commutator calculus

Secondary: 57M07: Topological methods in group theory

**Keywords**

Commutators free groups products of commutators

#### Citation

Sarkar, Sucharit. Commutators and squares in free groups. Algebr. Geom. Topol. 4 (2004), no. 1, 595--602. doi:10.2140/agt.2004.4.595. https://projecteuclid.org/euclid.agt/1513882489