## Illinois Journal of Mathematics

### Growing words in the free group on two generators

#### Abstract

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

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

Dates
First available in Project Euclid: 1 February 2013

https://projecteuclid.org/euclid.ijm/1359762395

Digital Object Identifier
doi:10.1215/ijm/1359762395

Mathematical Reviews number (MathSciNet)
MR3020689

Zentralblatt MATH identifier
1279.20037

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

#### Citation

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. https://projecteuclid.org/euclid.ijm/1359762395

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