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On some residual properties of the verbal embeddings of groups. (English) Zbl 1381.20029
Summary: We consider verbal embedding constructions preserving some residual properties for groups. An arbitrary residually finite countable group $$H$$ has a $$V$$-verbal embedding into a residually finite 2-generator group $$G$$ for any non-trivial word set $$V$$. If in addition $$H$$ is a residually soluble (residually nilpotent) group, then the group $$G$$ can be constructed also to be residually soluble (residually nilpotent). The analogs of this embedding also are true without the requirement about residual finiteness: Any residually soluble (residually nilpotent) countable group $$H$$ for any non-trivial word set $$V$$ has a $$V$$-verbal embedding into a residually soluble (residually nilpotent) 2-generator group $$G$$.
##### MSC:
 20E26 Residual properties and generalizations; residually finite groups 20E10 Quasivarieties and varieties of groups 20F19 Generalizations of solvable and nilpotent groups 20E22 Extensions, wreath products, and other compositions of groups 20F05 Generators, relations, and presentations of groups
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