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

Alan D. Logan

Research output: Contribution to journalArticle


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.

Original languageEnglish
Pages (from-to)116-152
Number of pages37
JournalAdvances in Mathematics
Early online date2 Jul 2019
Publication statusPublished - 7 Sep 2019



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

ASJC Scopus subject areas

  • Mathematics(all)

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