Profinite genus of HNN-extensions with finite associated subgroups

TL;DR

This paper investigates the classification of HNN-extensions with finite associated subgroups through their profinite completions, providing formulas and conditions for when these structures are uniquely determined by their profinite data.

Contribution

It offers explicit formulas for the number of isomorphism classes and the profinite genus of such HNN-extensions, advancing understanding of their profinite rigidity.

Findings

Formulas for isomorphism classes of HNN-extensions

Computed profinite genus of specific HNN-extensions

Identified conditions for profinite determination of HNN-extensions

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.