Extended Nielsen transformations and triviality of a group. (English. Russian original) Zbl 0573.20029

Math. Notes 35, 258-261 (1984); translation from Mat. Zametki 35, 491-495 (1984).
It was conjectured by J. J. Andrews and M. L. Curtis [Proc. Am. Math. Soc. 16, 192-195 (1965; Zbl 0131.38301) and Am. Math. Mon. 73, 21-28 (1966; Zbl 0135.044)] that if the trivial group has a finite balanced presentation \(<x_ 1,...,x_ n|\) \(r_ 1,...,r_ n>\) then \(\{r_ 1,...,r_ n\}\) can be transformed into \(\{x_ 1,...,x_ n\}\) by a sequence of transformations each of which is either an elementary Nielsen transformation or else consists of replacing some element by a conjugate. The author gives a proof of this conjecture in the variety of solvable groups.
Reviewer: G.Rosenberger


20F05 Generators, relations, and presentations of groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
55Q99 Homotopy groups
20F38 Other groups related to topology or analysis
55P99 Homotopy theory
Full Text: DOI


[1] S. Andrews and M. Curtis, ?Free groups and Handlebodies,? Proc. Am. Math. Soc.,16, 192-195 (1965).
[2] S. Andrews and M. Curtis, ?Extended Nielsen operators in free groups,? Am. Math. Mon.,73, 21-28 (1966). · Zbl 0135.04403
[3] R. Craggs, ?Free Heegaard diagrams and extended Nielsen transformations. I,? Illinois J. Math.,23, No. 1, 101-127 (1979). · Zbl 0404.57009
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