About the cyclically reduced product of words

TL;DR

This paper investigates the properties of the cyclically reduced product of words, revealing that while it lacks some classical algebraic properties, it still shares key features with free groups, impacting group theory conjectures.

Contribution

The paper provides a detailed study of the cyclically reduced product, establishing weaker algebraic properties and drawing parallels with free groups, which has implications for the Andrews-Curtis conjecture.

Findings

Weak forms of associativity and Latin square properties hold for the cyclically reduced product.

The set of cyclically reduced words with this product shares properties with free groups.

Insights may influence approaches to the Andrews-Curtis conjecture.

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.