On the word problem and the conjugacy problem for groups of the form $F/V(R)$

M. I. Anokhin

Matematičeskie zametki, Tome 61 (1997) no. 1, pp. 3-9

Résumé

Let $F$ be a free group with at most countable system $\mathfrak x$ of free generators, let $R$ be its normal subgroup recursively enumerable with respect to $\mathfrak x$, and let $\mathfrak V$ be a variety of groups that differs from $\mathfrak O$ and for which the corresponding verbal subgroup $V$ of the free group of countable rank is recursive. It is proved that the word problem in $F/V(R)$ is solvable if and only if this problem is solvable in $F/R$, and if $|\mathfrak x|\ge3$, then there exists an $R$ such, that the conjugacy problem in $F/R$ is solvable, but this problem is unsolvable in $F/V(R)$ for any Abelian variety $\mathfrak V\ne\mathfrak E$ (all algorithmic problems are regarded with respect to the images of $\mathfrak x$ under the corresponding natural epimorphisms).