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On conjectures of Andrews and Curtis. (English) Zbl 06852821
Summary: 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 J. J. Andrews and M. L. Curtis [Am. Math. Mon. 73, 21–28 (1966; Zbl 0135.04403)]. 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.
MSC:
20F05 Generators, relations, and presentations of groups
20F06 Cancellation theory of groups; application of van Kampen diagrams
57M20 Two-dimensional complexes (manifolds) (MSC2010)
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