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The intersection of the subgroups of finite $p$-index in a multiple HNN-extension of an infinite cyclic group

V. Metaftsis, D. Tsipa

TL;DR

The paper determines the intersection $(N_p)_\omega(G)$ of all normal subgroups of finite $p$-index in a multiple HNN-extension $G$ of the infinite cyclic group, thereby generalizing Moldavanskii’s Baumslag–Solitar results. It develops a framework using $p$-adic decompositions of the defining exponents, a holomorph-based embedding, and diophantine-analytic techniques to describe the residual finiteness properties of $G$. The main theorem specifies when $(N_p)_\omega(G)$ is the normal closure of a single power $a^{p^{\xi}}$ and, in the all-infinite case, when it is generated by a structured set of commutator-like elements; a corollary then gives exact necessary and sufficient conditions for $G$ to be residually finite $p$-group. These results advance understanding of residual finiteness in Generalized Baumslag–Solitar groups and multi-relator HNN-extensions, offering concrete criteria rooted in the $p$-adic behavior of the edge maps.

Abstract

Let $G$ be a multiple HNN-extension of an infinite cyclic group. We will calculate the intersection $(N_{p}){_ω}(G)$ of the normal subgroups of finite $p$-index in $G$ thus generalizing the result of Moldavanskii for Baumslag-Solitar groups. As a corollary we give necessary and sufficient conditions for $G$ to be residually finite $p$-group.

The intersection of the subgroups of finite $p$-index in a multiple HNN-extension of an infinite cyclic group

TL;DR

The paper determines the intersection of all normal subgroups of finite -index in a multiple HNN-extension of the infinite cyclic group, thereby generalizing Moldavanskii’s Baumslag–Solitar results. It develops a framework using -adic decompositions of the defining exponents, a holomorph-based embedding, and diophantine-analytic techniques to describe the residual finiteness properties of . The main theorem specifies when is the normal closure of a single power and, in the all-infinite case, when it is generated by a structured set of commutator-like elements; a corollary then gives exact necessary and sufficient conditions for to be residually finite -group. These results advance understanding of residual finiteness in Generalized Baumslag–Solitar groups and multi-relator HNN-extensions, offering concrete criteria rooted in the -adic behavior of the edge maps.

Abstract

Let be a multiple HNN-extension of an infinite cyclic group. We will calculate the intersection of the normal subgroups of finite -index in thus generalizing the result of Moldavanskii for Baumslag-Solitar groups. As a corollary we give necessary and sufficient conditions for to be residually finite -group.
Paper Structure (4 sections, 10 theorems, 24 equations)

This paper contains 4 sections, 10 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. Proof of the main Theorem
  4. Proof of the Corollary

Key Result

Theorem 1

[Moldavanskii] Let $G=BS(m,n)$, $p$ be a prime number and let $m=p^rm_1$ and $n=p^sn_1$ where $r,s\ge 0$ and $m_1,n_1$ are not divisible by $p$. Let also $d={\rm gcd}(m_1,n_1)$, $m_1=du$ and $n_1=dv$. Then

Theorems & Definitions (16)

  • Theorem
  • Corollary 1.1
  • Remark
  • Theorem
  • Corollary
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Theorem
  • Lemma 2.3
  • ...and 6 more