We obtain both necessary and sufficient conditions for the free product of two groups with normal amalgamated subgroups to be a residually C-group, where C is a root class of groups, which must be homomorphically closed in most cases.

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On the root-class residuality of generalized free products with a normal amalgamation

Given a class K of groups, we prove that the free product of a K -group A and a residually K -group B with amalgamated subgroup which is a retract of B is a residually K -group. We also obtain a sufficient condition for the root-class residuality of a generalized free product of two residually K -groups with amalgamated subgroup which is a retract of one of the factors.

Sufficient conditions for the root-class residuality of certain generalized free products

Given a root class $\mathscr{K}$
 of groups, we prove that the tree product of residually $\mathscr{K}$
 -groups with amalgamated retracts is a residually $\mathscr{K}$
 -group. This yields a criterion for the $\mathscr{K}$
 -residuality of Artin and Coxeter groups with tree structure. We also prove that the HNN-extension X of a residually $\mathscr{K}$
 -group B is a residually $\mathscr{K}$
 -group provided that the associated subgroups of X are retracts in B and $\mathscr{K}$
 contains at least one nonperiodic group.

The Root Class Residuality of the Tree Product of Groups with Amalgamated Retracts

Let R be a root class of groups which is closed with respect to passage to quotient groups and contains at least one nonidentity group. A criterion for the R-residuality of an HNN-extension whose associated subgroups are cyclic and belong to the center of the base group is obtained.

Root class residuality of HNN-extensions with central cyclic associated subgroups

Let $${\cal K}$$ be a root class of groups closed under taking quotient groups, G be a free product of groups A and B with amalgamated subgroups H and K. Let also H be normal in A, K be normal in B, and AutG(H) denote the set of automorphisms of H induced by all inner automorphisms of G. We prove a criterion for G to be residually a $${\cal K}$$-group provided AutG(H) is an abelian group or it satisfies some other conditions. We apply this result in the cases when A and B are bounded nilpotent groups or A/H, B/K ∈ $${\cal K}$$.

On the Root-Class Residuality of Certain Free Products of Groups with Normal Amalgamated Subgroups

HNN-extensions of a group with central associated subgroups are considered. Several conditions are obtained for the approximability of such HNN-extensions by a class of groups which have the root property and is closed with respect to factorization.

D. V. Gol’tsov

Papers

Approximability of HNN-extensions with central associated subgroups by a root class of groups