Certain residual properties of HNN-extensions with normal associated subgroups

TL;DR

The paper investigates when an HNN-extension $E=\langle B,t; t^{-1}Ht=K, \varphi\rangle$ with normal $H,K$ and $L=H\cap K$ φ-invariant is residually a $\mathcal{C}$-group for a root class $\mathcal{C}$ closed under subgroups, quotients, and wreath products. It develops a two-level filtration approach: first, obtain onto maps to $\mathcal{C}$-groups that are injective on the base, then construct large families of such quotients via free constructions to derive residual $\mathcal{C}$-ness even when $B$ itself need not lie in $\mathcal{C}$. The main contributions are necessary and sufficient criteria linking residual $\mathcal{C}$-ness of $E$ to automorphism subgroups $\mathfrak{U}=\mathrm{Aut}_{B}(H)$ and $\mathfrak{V}=\mathrm{Aut}_{\mathbb{E}}(L)$, along with refined results under additional assumptions (e.g., finiteness of $H,K$, quasi-regularity, and periodic root-class settings). These results generalize and extend classical residual finiteness and residual $p$-finiteness criteria for HNN-extensions, offering a unified framework for a broad class of residual properties in free constructions of groups.

Abstract

Let $\mathbb{E}$ be the HNN-extension of a group $B$ with subgroups $H$ and $K$ associated according to an isomorphism $\varphi\colon H \to K$. Suppose that $H$ and $K$ are normal in $B$ and $(H \cap K)\varphi = H \cap K$. 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 $\mathbb{E}$.

Theorems & Definitions (52)

• Theorem 1
• Theorem 2
• Corollary 1
• Theorem 3
• Theorem 4
• Example
• Theorem 5
• Theorem 6
• Corollary 2
• Proposition 3.1