Weinbaum’s conjecture on unique subwords of nonperiodic words

Duncan, Andrew J.; Howie, James

doi:10.1090/S0002-9939-1992-1110541-5

Weinbaum’s conjecture on unique subwords of nonperiodic words
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by Andrew J. Duncan and James Howie PDF

Proc. Amer. Math. Soc. 115 (1992), 947-954 Request permission

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.

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