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Profinite genus of HNN-extensions with finite associated subgroups

V. R. de Bessa, A. L. P. Porto, P. A. Zalesskii

TL;DR

This paper analyzes the profinite genus of HNN-extensions over finite subgroups, introducing $\widehat{OE}$-groups and establishing a profinite Bass–Serre framework to count isomorphism classes of both abstract and profinite HNN-extensions. It derives explicit formulas for the numbers of isomorphism classes, provides genus computations in the fixed/not-fixed setting, and proves when an HNN-extension is determined by its profinite completion (genus 1) across various scenarios and base-group families. It also extends these results to the broader class $\mathcal{A}$ of finitely generated residually finite accessible groups, giving summation formulas over base representatives and compatibility conditions. The results yield concrete genus bounds and exact counts in many natural cases, including when the base is a $\widehat{OE}$-group and when associated subgroups are conjugate or not, thereby clarifying when profinite data suffices to recover the abstract structure.

Abstract

We study the profinite genus of HNN-extensions whose associated subgroups are finite. We give precise formulas for the number of isomorphism classes of HNN(G,H,K,t,f) and of its profinite completion and compute the profinite genus of such an HNN-extension HNN(G,H,K,t,f). We also list various situations when HNN(G,H,K,t,f) is determined by its profinite completion.

Profinite genus of HNN-extensions with finite associated subgroups

TL;DR

This paper analyzes the profinite genus of HNN-extensions over finite subgroups, introducing -groups and establishing a profinite Bass–Serre framework to count isomorphism classes of both abstract and profinite HNN-extensions. It derives explicit formulas for the numbers of isomorphism classes, provides genus computations in the fixed/not-fixed setting, and proves when an HNN-extension is determined by its profinite completion (genus 1) across various scenarios and base-group families. It also extends these results to the broader class of finitely generated residually finite accessible groups, giving summation formulas over base representatives and compatibility conditions. The results yield concrete genus bounds and exact counts in many natural cases, including when the base is a -group and when associated subgroups are conjugate or not, thereby clarifying when profinite data suffices to recover the abstract structure.

Abstract

We study the profinite genus of HNN-extensions whose associated subgroups are finite. We give precise formulas for the number of isomorphism classes of HNN(G,H,K,t,f) and of its profinite completion and compute the profinite genus of such an HNN-extension HNN(G,H,K,t,f). We also list various situations when HNN(G,H,K,t,f) is determined by its profinite completion.
Paper Structure (24 sections, 66 theorems, 176 equations)

This paper contains 24 sections, 66 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Notations and terminologies
  3. Preliminary Results
  4. Basics of profinite trees
  5. Normalizers
  6. OE-groups
  7. General results
  8. Isomorphism problem for abstract HNN-extensions
  9. The number of isomorphism classes of abstract HNN-extensions
  10. Normal HNN-extension
  11. Fixed associated subgroups
  12. Not fixed associated subgroups
  13. Examples
  14. Profinite HNN-extensions
  15. Isomorphism problem for profinite HNN-extensions
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $G_1$ be an ${\rm OE}$-group and be abstract HNN-extensions with fixed finite associated subgroups $H,K$ of $G_1$. Then $G$ and $B$ are isomorphic if and only if $f$ and $f_1$ belong to the same $\overline\Gamma_{HK}$-orbit in ${\rm Iso}(H,K)$.

Theorems & Definitions (137)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 127 more