Twisted associativity of the cyclically reduced product of words, part 2

Abstract

The cyclically reduced product of two words $u,v$, denoted $u\ast v$, is the cyclically reduced form of the concatenation of $u$ by $v$. This product is not associative. Recently S. V. Ivanov has proved that the Andrews-Curtis conjecture can be restated in terms of the cyclically reduced product and cyclic permutations instead of the reduced product and conjugations. In a previous paper we have proved that $\ast$ verifies generalizations of properties of the product in the free group. In another previous paper we have proved that $\ast$ verifies a generalized version of the associativity property in a special case. In the present paper we prove that a more general version of the associativity property holds for $\ast$ in the general case.

arXiv comment68 pages, 4 figures. Author's note: some text overlaps with two previous papers of the author (arXiv:1908.07435, arXiv:1909.04863) in order to maintain the exposition self-contained. arXiv admin note: text overlap with arXiv:1908.07435

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• About the cyclically reduced product of words