Illinois Journal of Mathematics

Growing words in the free group on two generators

Bobbe Cooper and Eric Rowland

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This paper is concerned with minimal-length representatives of equivalence classes of words in $F_2$ under Aut $F_2$. We give a simple inequality characterizing words of minimal length in their equivalence class. We consider an operation that “grows” words from other words, increasing the length, and we study root words—minimal words that cannot be grown from other minimal words. Root words are “as minimal as possible” in the sense that their characterization is the boundary case of the minimality inequality. The property of being a root word is respected by equivalence classes, and the length of each root word is divisible by 4.

Article information

Illinois J. Math., Volume 55, Number 2 (2011), 417-426.

First available in Project Euclid: 1 February 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20E05: Free nonabelian groups 68R15: Combinatorics on words


Cooper, Bobbe; Rowland, Eric. Growing words in the free group on two generators. Illinois J. Math. 55 (2011), no. 2, 417--426. doi:10.1215/ijm/1359762395.

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