On the Andrews-Curtis hypothesis.

*(Russian)*Zbl 0601.20035
Monogenic functions and mappings, Collect. sci. Works, Kiev 1982, 52-58 (1982).

[For the entire collection see Zbl 0506.00009.]

Nielsen’s theorem that the automorphism group \(\operatorname{Aut}(F_ k)\) of a free group of rank k can be finitely generated by elementary automorphisms corresponding to Nielsen operations can be restated as follows [see J. J. Andrews and M. L. Curtis, Proc. Am. Math. Soc. 16, 192-195 (1965; Zbl 0131.38301)]: if \(X=<x_ 1,...,x_ k>\) and \(Y=<y_ 1,...,y_ k>\) are free bases of \(F_ k\), then there is a finite sequence of operations of types (1)-(3) that will change X into Y.

(1) Permute the elements in the given set X;

(2) Replace \(x_ 1\) by its inverse \(x_ 1^{-1};\)

(3) Replace \(x_ 1\) by the product \(x_ 1x_ 2;\)

(4) Replace \(x_ 1\) by any conjugate \(g^{-1}x_ 1g\), where \(g\in F_ k.\)

Conjecture. [Andrews and Curtis, op. cit.]. If \(R=<r_ 1,...,r_ k>\subseteq F_ k\) has normal closure equal to \(F_ k\) (i.e., if \(<X| R>\) is a presentation of the trivial group), then R may be changed into X by a finite sequence of operations of types (1)-(4).

The author of the paper under review establishes the following partial result for ”short” basis elements. Theorem. Let \(U=(u_ 1,...,u_ m)\subseteq F_ k\) be a subset of \(F_ k\) whose normal closure is the entire group \(F_ k\). If \(r\leq 4\) and if \(\alpha =\prod^{r}_{i=1}g_ i^{-1}v_ ig_ i\), where \(g_ i\in F_ k\), \(v_ i\in U^{+1}\), then U can be transformed by operations (1)-(4) into \(U'=(\alpha,u_ 1,...,\tilde u_ j,...,u_ m)\)- with a \(u_ j\) omitted.

Although a general theorem is not proved, the author offers the following main result. Recall that an element of a free group F is primitive if the quotient by its normal closure is a free group of rank rank(F)-1.

Theorem 2. A necessary and sufficient condition for the generalized Andrews-Curtis hypothesis to hold is that there exists a primitive element is some subgroup of \(F_ k\) generated by some set obtained from \(R=<r_ 1,...,r_ m>\) by operations of types (1)-(4). - The generalized Andrews-Curtis hypothesis allows any number of normal generators (necessarily \(m\geq k)\) and requires that R be transformed into (X,1,1,...,1), where X is any basis of \(F_ k\) (with k elements).

Nielsen’s theorem that the automorphism group \(\operatorname{Aut}(F_ k)\) of a free group of rank k can be finitely generated by elementary automorphisms corresponding to Nielsen operations can be restated as follows [see J. J. Andrews and M. L. Curtis, Proc. Am. Math. Soc. 16, 192-195 (1965; Zbl 0131.38301)]: if \(X=<x_ 1,...,x_ k>\) and \(Y=<y_ 1,...,y_ k>\) are free bases of \(F_ k\), then there is a finite sequence of operations of types (1)-(3) that will change X into Y.

(1) Permute the elements in the given set X;

(2) Replace \(x_ 1\) by its inverse \(x_ 1^{-1};\)

(3) Replace \(x_ 1\) by the product \(x_ 1x_ 2;\)

(4) Replace \(x_ 1\) by any conjugate \(g^{-1}x_ 1g\), where \(g\in F_ k.\)

Conjecture. [Andrews and Curtis, op. cit.]. If \(R=<r_ 1,...,r_ k>\subseteq F_ k\) has normal closure equal to \(F_ k\) (i.e., if \(<X| R>\) is a presentation of the trivial group), then R may be changed into X by a finite sequence of operations of types (1)-(4).

The author of the paper under review establishes the following partial result for ”short” basis elements. Theorem. Let \(U=(u_ 1,...,u_ m)\subseteq F_ k\) be a subset of \(F_ k\) whose normal closure is the entire group \(F_ k\). If \(r\leq 4\) and if \(\alpha =\prod^{r}_{i=1}g_ i^{-1}v_ ig_ i\), where \(g_ i\in F_ k\), \(v_ i\in U^{+1}\), then U can be transformed by operations (1)-(4) into \(U'=(\alpha,u_ 1,...,\tilde u_ j,...,u_ m)\)- with a \(u_ j\) omitted.

Although a general theorem is not proved, the author offers the following main result. Recall that an element of a free group F is primitive if the quotient by its normal closure is a free group of rank rank(F)-1.

Theorem 2. A necessary and sufficient condition for the generalized Andrews-Curtis hypothesis to hold is that there exists a primitive element is some subgroup of \(F_ k\) generated by some set obtained from \(R=<r_ 1,...,r_ m>\) by operations of types (1)-(4). - The generalized Andrews-Curtis hypothesis allows any number of normal generators (necessarily \(m\geq k)\) and requires that R be transformed into (X,1,1,...,1), where X is any basis of \(F_ k\) (with k elements).

Reviewer: M.Garzon