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

Alan D. Logan

doi:10.1016/j.aim.2019.06.009

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

Alan D. Logan

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

287 Downloads (Pure)

Abstract

Fix an equilateral triangle group T_i=〈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 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
Pages (from-to)	116-152
Number of pages	37
Journal	Advances in Mathematics
Volume	353
Early online date	2 Jul 2019
DOIs	https://doi.org/10.1016/j.aim.2019.06.009
Publication status	Published - 7 Sept 2019

Keywords

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

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1016/j.aim.2019.06.009

Download pdfAccepted author manuscript, 514 KBLicence: CC BY-NC-ND

Cite this

@article{a6b48df5f499420782ca7b70a94d8331,

title = "Every group is the outer automorphism group of an HNN-extension of a fixed triangle group",

abstract = "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.",

keywords = "Automorphisms of free groups, HNN-extensions, Outer automorphism groups, Small cancellation theory, Triangle groups",

author = "Logan, \{Alan D.\}",

year = "2019",

month = sep,

day = "7",

doi = "10.1016/j.aim.2019.06.009",

language = "English",

volume = "353",

pages = "116--152",

journal = "Advances in Mathematics",

issn = "0001-8708",

publisher = "Academic Press Inc.",

}

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 - https://www.scopus.com/pages/publications/85068121180

U2 - 10.1016/j.aim.2019.06.009

DO - 10.1016/j.aim.2019.06.009

M3 - Article

AN - SCOPUS:85068121180

SN - 0001-8708

VL - 353

SP - 116

EP - 152

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this