### Abstract

Fix an equilateral triangle group T_{i}=〈a,b;a^{i},b^{i},(ab)^{i}〉 with i≥6 arbitrary. Our main result is: for every presentation P of every countable group Q there exists an HNN-extension T_{P} of T_{i} such that Out(T_{P})≅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.

Original language | English |
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Pages (from-to) | 116-152 |

Number of pages | 37 |

Journal | Advances in Mathematics |

Volume | 353 |

Early online date | 2 Jul 2019 |

DOIs | |

Publication status | Published - 7 Sep 2019 |

### Keywords

- Automorphisms of free groups
- HNN-extensions
- Outer automorphism groups
- Small cancellation theory
- Triangle groups

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Logan, A. D. (2019). Every group is the outer automorphism group of an HNN-extension of a fixed triangle group.

*Advances in Mathematics*,*353*, 116-152. https://doi.org/10.1016/j.aim.2019.06.009