Certain residual properties of HNN-extensions with normal associated subgroups

Abstract Let 𝔼 be the HNN-extension of a group 𝐵 with subgroups 𝐻 and 𝐾 associated by an isomorphism <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>φ</m:mi> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mrow> <m:mi>H</m:mi> <m:mo stretchy="false">→</m:mo> <m:mi>K</m:mi> </m:mrow> </m:mrow> </m:math> \varphi\colon H\to K . Suppose that 𝐻 and 𝐾 are normal in 𝐵 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>H</m:mi> <m:mo>∩</m:mo> <m:mi>K</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mi>φ</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>H</m:mi> <m:mo>∩</m:mo> <m:mi>K</m:mi> </m:mrow> </m:mrow> </m:math> (H\cap K)\varphi=H\cap K . Under these assumptions, we prove necessary and sufficient conditions for 𝔼 to be residually a 𝒞-group, where 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 𝔼.