Certain residual properties of HNN-extensions with normal associated subgroups

Abstract

Let $\mathbb{E}$ be the HNN-extension of a group $B$ with subgroups $H$ and $K$ associated according to an isomorphism $\phi :H\to K$. Suppose that $H$ and $K$ are normal in $B$ and $(H\cap K)\phi =H\cap K$. Under these assumptions, we prove necessary and sufficient conditions for $\mathbb{E}$ to be residually a $\mathcal{C}$-group, where $\mathcal{C}$ is a class of groups closed under taking subgroups, quotient groups, and unrestricted wreath products. Among other things, these conditions give new facts on the residual finiteness and the residual $p$-finiteness of the group $\mathbb{E}$.

arXiv comment20 pages; the English version of the previously published Russian original