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Embedding of groups and quadratic equations over groups. (English) Zbl 1372.20032

Let \(G\) be a countable (possibly finite) group. By a well-known theorem of G. Higman et al. [J. Lond. Math. Soc. 24, 247–254 (1950; Zbl 0034.30101)], the group \(G\) can be embedded into some 2-generator group \(H\). The paper under review is devoted to proving (Theorem 1.1) that for each integer \(n\geq 2\) this can be done in such a way that every quadratic cyclically reduced word equation \(W\) of length at most n has a solution in \(G\) if and only if it has one in \(H\).
We need to explain some of these terms. The word \(W\) here involves variables, that is, free generators in some external free group and the length of \(W\) here means the total number of occurrences of variables in \(W\). Also \(W\) is quadratic if for each variable \(x\) appearing in \(W\) the total number of occurrences of \(x\) in \(W\) either as an \(x\) or as an \(x^{-1}\) is 2.
The author also proves that the group \(H\) and the embedding of \(G\) into \(H\) can be constructed so as to have a number of other rather technical properties, see Theorem 1.2.

MSC:

20F05 Generators, relations, and presentations of groups
20F70 Algebraic geometry over groups; equations over groups
20F06 Cancellation theory of groups; application of van Kampen diagrams

Citations:

Zbl 0034.30101
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Full Text: arXiv Euclid