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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).
Reviewer: M.Garzon
##### MSC:
 20F05 Generators, relations, and presentations of groups 20E05 Free nonabelian groups