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.
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. MR0340420 0288.20027 10.1112/jlms/s2-8.2.254 P. J. Higgins and R. C. Lyndon, Equivalence of elements under automorphisms of a free group, J. Lond. Math. Soc. 8 (1974), 254–258. MR0340420 0288.20027 10.1112/jlms/s2-8.2.254
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. MR2077762 10.1090/conm/349/06360 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. MR2077762 10.1090/conm/349/06360
D. Lee, Counting words of minimum length in an automorphic orbit, J. Algebra 301 (2006), 35–58. MR2230319 10.1016/j.jalgebra.2006.04.012 1100.20028 D. Lee, Counting words of minimum length in an automorphic orbit, J. Algebra 301 (2006), 35–58. MR2230319 10.1016/j.jalgebra.2006.04.012 1100.20028
D. Lee, A tighter bound for the number of words of minimum length in an automorphic orbit, J. Algebra 305 (2006), 1093–1101. MR2266870 10.1016/j.jalgebra.2006.03.038 D. Lee, A tighter bound for the number of words of minimum length in an automorphic orbit, J. Algebra 305 (2006), 1093–1101. MR2266870 10.1016/j.jalgebra.2006.03.038
A. G. Myasnikov and V. Shpilrain, Automorphic orbits in free groups, J. Algebra 269 (2003), 18–27. MR2015300 10.1016/S0021-8693(03)00339-9 1035.20019 A. G. Myasnikov and V. Shpilrain, Automorphic orbits in free groups, J. Algebra 269 (2003), 18–27. MR2015300 10.1016/S0021-8693(03)00339-9 1035.20019
E. S. Rapaport, On free groups and their automorphisms, Acta Mathematica 99 (1958), 139–163. MR0131452 0080.01705 10.1007/BF02392424 E. S. Rapaport, On free groups and their automorphisms, Acta Mathematica 99 (1958), 139–163. MR0131452 0080.01705 10.1007/BF02392424
C. M. Sanchez, Minimal words in the free group of rank two, J. Pure Appl. Algebra 17 (1980), 333–337. MR0579093 10.1016/0022-4049(80)90055-9 0438.20016 C. M. Sanchez, Minimal words in the free group of rank two, J. Pure Appl. Algebra 17 (1980), 333–337. MR0579093 10.1016/0022-4049(80)90055-9 0438.20016
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. http://www.math.oregonstate.edu/~math_reu/proceedings/REU_Proceedings/Proceedings1998/1998Virning.pdf 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. 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. MR1575455 10.1112/plms/s2-41.1.48 62.0079.04 J. H. C. Whitehead, On certain sets of elements in a free group, Proc. Lond. Math. Soc. 41 (1936), 48–56. MR1575455 10.1112/plms/s2-41.1.48 62.0079.04
J. H. C. Whitehead, On equivalent sets of elements in a free group, Ann. of Math. 37 (1936), 782–800. MR1503309 10.2307/1968618 62.1093.04 J. H. C. Whitehead, On equivalent sets of elements in a free group, Ann. of Math. 37 (1936), 782–800. MR1503309 10.2307/1968618 62.1093.04
