Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Weinbaum's conjecture on unique subwords of nonperiodic words


Authors: Andrew J. Duncan and James Howie
Journal: Proc. Amer. Math. Soc. 115 (1992), 947-954
MSC: Primary 20F05; Secondary 20F10
DOI: https://doi.org/10.1090/S0002-9939-1992-1110541-5
MathSciNet review: 1110541
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Following a conjecture of Weinbaum, we show that every nonperiodic word $ W$ of length at least 2 in a free group has a cyclic permutation of the form UV, where each of $ U$ and $ V$ occur precisely once as a cyclic subword of $ W$ and neither occurs as a cyclic subword of $ {W^{ - 1}}$. In fact, we prove a somewhat stronger version of this result and also give a number of applications to one-relator products of groups.


References [Enhancements On Off] (What's this?)

  • [1] S. D. Brodskii, Equations over groups and groups with a single defining relator, Siberian Math. J. 25 (1984), 235-251. MR 741011 (86e:20026)
  • [2] R. G. Burns and V. W. D. Hale, A note on group rings of certain torsion-free groups, Canad. Math. Bull. 15 (1972), 441-445. MR 0310046 (46:9149)
  • [3] J. Howie, On locally indicable groups, Math. Z. 180 (1982), 445-461. MR 667000 (84b:20036)
  • [4] -, The quotient of a free product of groups by a single high-powered relator, I. Pictures, Fifth and higher powers, Proc. London Math. Soc. (3) 59 (1989), 507-540. MR 1014869 (90j:20055)
  • [5] -, The quotient of a free product of groups by a single high-powered relator, II. Fourth powers, Proc. London Math. Soc. (3) 61 (1990), 33-62. MR 1051098 (91d:20033)
  • [6] -, The quotient of a free product of groups by a single high-powered relator, III. The word problem, Proc. London Math. Soc. (3) 62 (1991), 590-606. MR 1095234 (92d:20044)
  • [7] S. J. Pride, Small cancellation conditions satisfied by one-relator groups, Math. Z. 184 (1983), 283-286. MR 716277 (85e:20028)
  • [8] C. M. Weinbaum, On relators and diagrams for groups with a single defining relator, Illinois J. Math. 16 (1972), 308-322. MR 0297849 (45:6901)
  • [9] -, Unique subwords in nonperiodic words, Proc. Amer. Math. Soc. 109 (1990), 615-619. MR 1017852 (90k:20097)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 20F05, 20F10

Retrieve articles in all journals with MSC: 20F05, 20F10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1110541-5
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society