Every group is the outer automorphism group of an HNN-extension of a fixed triangle group

TL;DR

The paper demonstrates that every countable group can be realized as the outer automorphism group of an explicitly constructed HNN-extension of a fixed triangle group, expanding understanding of automorphism groups in geometric group theory.

Contribution

It introduces a method to realize any countable group as an outer automorphism group of an HNN-extension of a fixed triangle group, with explicit constructions and new subgroup recognition techniques.

Findings

Every countable group is the outer automorphism group of some HNN-extension of a fixed triangle group.

Introduces a method for recognizing malnormal subgroups in small cancellation groups.

Defines the concept of 'malcharacteristic' subgroups for subgroup analysis.

Abstract

Fix an equilateral triangle group $T_i=\langle a, b; a^i, b^i, (ab)^i\rangle$ with $i\geq6$ arbitrary. Our main result is: for every presentation $\mathcal{P}$ of every countable group $Q$ there exists an HNN-extension $T_{\mathcal{P}}$ of $T_i$ such that $\operatorname{Out}(T_{\mathcal{P}})\cong Q$. We construct the HNN-extensions explicitly, and examples are given. The class of groups constructed have nice categorical and residual properties. In order to prove our main result we give a method for recognising malnormal subgroups of small cancellation groups, and we introduce the concept of "malcharacteristic" subgroups.