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 of length at least 2 in a free group has a cyclic permutation of the form *UV*, where each of and occur precisely once as a cyclic subword of and neither occurs as a cyclic subword of . In fact, we prove a somewhat stronger version of this result and also give a number of applications to one-relator products of groups.

**[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)**

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