On Some Residual Properties of the Verbal Embeddings of Groups

Abstract

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 $G$.