Abstract:
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.
Keywords:tree product of groups, HNN-extension, Artin group, Coxeter group, root class residuality, residual finiteness, residual $p$-finiteness, residual solubility.
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