About the cyclically reduced product of words

Abstract

The cyclically reduced product of two words is the cyclically reduced form of the concatenation of the two words. While the reduced form of such a concatenation (which is the product of the free group) verifies many basic properties like for example associativity, the same is not true for the cyclically reduced product which has been very little studied in the literature. Recently Sergei Ivanov has proved that the Andrews-Curtis conjecture (stated in 1965 and still not solved) is equivalent to a formulation where the reduced product is replaced by the cyclically reduced product (and the conjugations replaced by cyclic permutations). In this paper we study properties of the cyclically reduced product $*$ and of the set of cyclically reduced words $\hat{\mathcal{F}(X)}$ equipped with $*$. In particular we find that even if $*$ is not commutative nor verifies the Latin square property, weaker versions of these properties hold true. We also show that $\hat{\mathcal{F}(X)}$ equipped with $*$ and with cyclic permutations enjoys similar properties as the free group equipped with the reduced product and conjugations.