## 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

### MSC:

 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

### Citations:

Zbl 0131.38301; Zbl 0135.044
Full Text:

### References:

 [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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