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7.24 (1980)

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We say that a group is sparse if the variety generated by it has at most countably many subvarieties. Does there exist a finitely generated sparse group that has undecidable word problem?

Contributor: A. V. Kuznetsov

Progress

Yes, such groups do exist; for example, a free group of $\mathfrak{N}_3\mathfrak{A}$. By (A. N. Krasil’nikov, Math. USSR–Izv., 37 (1991), 539–553) every subvariety of $\mathfrak{N}_3\mathfrak{A}$ has a finite basis for its laws; hence there are only countably many of them. It has undecidable word problem by (O. G. Kharlampovich, Sov. Math., 32, no. 11 (1988), 136–140).

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