Some embedding theorems and undecidability questions for groups. (English) Zbl 0843.20027

Gordon, C. McA.

Duncan, Andrew J. (ed.) et al., Combinatorial and geometric group theory. Proceedings of a workshop held at Heriot-Watt University, Edinburgh, GB, spring of 1993. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 204, 105-110 (1995).

Simplified direct constructive proofs, using generalized free products of groups, of some known embedding theorems for groups are presented, for example of the theorem of A. P. Goryushkin [Mat. Zametki 16, 231-235 (1974; Zbl 0306.20040)] that every countable group can be embedded in a 2-generator simple group. {The paper by P. E. Schupp [J. Lond. Math. Soc., II. Ser. 13, 90-94 (1976; Zbl 0363.20026)], which the author quotes for this result, too, contains the refinement that the generators of the embedding group can be chosen to have orders 2 and 3, respectively, which is not considered by the author.} The author then goes on to prove the algorithmic unsolvability of the question whether a finitely presented group has trivial Schur multiplicator, and that of the question what the deficiency of a finitely presented group is, and finally that of the question whether a finitely presented group is a higher-dimensional knot group. The use of small cancellation theory is avoided in the author’s approach.
For the entire collection see [Zbl 0830.00030].

Reviewer: B.H.Neumann (Canberra)

MSC:

20F10	Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20E06	Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F05	Generators, relations, and presentations of groups
20E07	Subgroup theorems; subgroup growth

Keywords:

generalized free products; embedding theorems for groups; countable groups; 2-generator simple groups; algorithmic unsolvability; deficiency; finitely presented groups