### 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.

Language | English |
---|---|

Pages | 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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

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

}

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

**Every group is the outer automorphism group of an HNN-extension of a fixed triangle group.** / Logan, Alan D.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Every group is the outer automorphism group of an HNN-extension of a fixed triangle group

AU - Logan, Alan D.

PY - 2019/9/7

Y1 - 2019/9/7

N2 - Fix an equilateral triangle group Ti=〈a,b;ai,bi,(ab)i〉 with i≥6 arbitrary. Our main result is: for every presentation P of every countable group Q there exists an HNN-extension TP of Ti such that Out(TP)≅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.

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

KW - Automorphisms of free groups

KW - HNN-extensions

KW - Outer automorphism groups

KW - Small cancellation theory

KW - Triangle groups

UR - http://www.scopus.com/inward/record.url?scp=85068121180&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2019.06.009

DO - 10.1016/j.aim.2019.06.009

M3 - Article

VL - 353

SP - 116

EP - 152

JO - Advances in Mathematics

T2 - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -