On Conjectures of Andrews and Curtis

Abstract

It is shown that the original Andrews-Curtis conjecture on balanced presentations of the trivial group is equivalent to its "cyclic" version in which, in place of arbitrary conjugations, one can use only cyclic permutations. This, in particular, proves a satellite conjecture of Andrews and Curtis [2] made in 1966. We also consider a more restrictive "cancellative" version of the cyclic Andrews-Curtis conjecture with and without stabilizations and show that the restriction does not change the Andrews-Curtis conjecture when stabilizations are allowed. On the other hand, the restriction makes the conjecture false when stabilizations are not allowed.

Twisted associativity of the cyclically reduced product of words, part 1

“…As the latter example shows, it appears that the structure of F (X) equipped with the cyclically reduced product and with cyclic permutations enjoys similar properties as those enjoyed by the free group equipped with the reduced product and conjugations. This was hinted at in the papers [10], [11], [4], [5]. In [12] we have explored further this fact; in particular we have proved that for words u and v the cyclically reduced product u * v is a cyclic permutation of v * u and the identity among relations that follows from this fact is a generalization of the identity among relations that follows from the fact that in the free group u • v is a conjugate of v • u.…”

“…The cyclically reduced product has applications to the Andrews-Curtis conjecture: in [4] and [5] S. V. Ivanov has proved that the conjecture (with and without stabilizations) is true if and only if in the definition of the conjecture we replace the operations of reduced product and conjugations with the cyclically reduced product and cyclic permutations. The importance of this result stems from the fact that while there are infinitely many conjugates of one word, there are only finitely many cyclic permutations, thus making much easier the search of Andrews-Curtis trivializations by enumerations of relators, like for example the approaches used in [1] or [9].…”

About the cyclically reduced product of words

“…In the above mentioned paper and in [6] S. V. Ivanov proved that the b A picture is a sort of dual of a van Kampen diagram, see Ch. 2 of [14] c We observe that the previous hypothesis is not restrictive because if X = X1 ∪ X2 with the elements of X1 or their inverses occurring in at least one basic relator and the elements of X2 not occurring, then the group presented by X|R is the free product of G1 times F(X2), where G1 is the group presented by X1|R .…”

“…But things have changed recently because in two papers of 2006 [5] and of 2018 [6] S. V. Ivanov has proved an extremely interesting result concerning the Andrews-Curtis conjecture: the conjecture (with and without stabilizations) is true if and only if in the definition of the conjecture we replace the operations of reduced product and conjugations with the cyclically reduced product and the cyclic permutations.…”

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

“…The cyclically reduced product has applications to the Andrews-Curtis conjecture: in [4] and [5] S. V. Ivanov has proved that the conjecture (with and without stabilizations) is true if and only if in the definition of the conjecture we replace the operations of reduced product and conjugations with the cyclically reduced product and cyclic permutations. The importance of this result stems from the fact that while there are infinitely many conjugates of one word, there are only finitely many cyclic permutations, thus making much easier the search of Andrews-Curtis trivializations by enumerations of relators, like for example the approaches used in [1] or [7].…”