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

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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

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10.1016/j.aim.2019.06.009

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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.",

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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.

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

Abstract

Keywords

ASJC Scopus subject areas

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