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

Abstract

Fix an equilateral triangle group $T_i=⟨a,b;a^i,b^i,(ab{)}^i⟩$ with $i\ge 6$ arbitrary. Our main result is: for every presentation $\mathcal{P}$ of every countable group $Q$ there exists an HNN-extension $T_{\mathcal{P}}$ of $T_i$ such that $\mathrm{Out}(T_{\mathcal{P}})\cong Q$. 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.

arXiv comment39 pages. Final version, to appear in Advances in Mathematics