Illinois Journal of Mathematics

Growing words in the free group on two generators

Bobbe Cooper and Eric Rowland

Full-text: Open access

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

Permanent link to this document
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


Export citation

References

  • P. J. Higgins and R. C. Lyndon, Equivalence of elements under automorphisms of a free group, J. Lond. Math. Soc. 8 (1974), 254–258.
  • B. Khan, The structure of automorphic conjugacy in the free group of rank two, Computational and Experimental Group Theory, Contemporary Mathematics, vol. 349, American Mathematical Society, Providence, RI, 2004, pp. 115–196.
  • D. Lee, Counting words of minimum length in an automorphic orbit, J. Algebra 301 (2006), 35–58.
  • D. Lee, A tighter bound for the number of words of minimum length in an automorphic orbit, J. Algebra 305 (2006), 1093–1101.
  • A. G. Myasnikov and V. Shpilrain, Automorphic orbits in free groups, J. Algebra 269 (2003), 18–27.
  • E. S. Rapaport, On free groups and their automorphisms, Acta Mathematica 99 (1958), 139–163.
  • C. M. Sanchez, Minimal words in the free group of rank two, J. Pure Appl. Algebra 17 (1980), 333–337.
  • R. Virnig, Whitehead automorphisms and equivalent words, Proceedings of the research experience for undergraduates program in mathematics, Oregon State University, Corvallis, OR, 1998, pp. 125–167; available at http://www.math.oregonstate.edu/~math_reu/proceedings/REU_Proceedings/Proceedings1998/1998Virning.pdf.
  • J. H. C. Whitehead, On certain sets of elements in a free group, Proc. Lond. Math. Soc. 41 (1936), 48–56.
  • J. H. C. Whitehead, On equivalent sets of elements in a free group, Ann. of Math. 37 (1936), 782–800.