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

Fix an equilateral triangle group $T_i=\langle a, b; a^i, b^i, (ab)^i\rangle$ with $i\geq6$ 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 $\operatorname{Out}(T_{\mathcal{P}})\cong Q$. 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.Comment: 39 pages. Final version, to appear in Advances in Mathematic

Linear and projective boundaries in HNN-extensions and distortion phenomena

Linear and projective boundaries of Cayley graphs were introduced in~\cite{kst} as quasi-isometry invariant boundaries of finitely generated groups. They consist of forward orbits $g^\infty=\{g^i: i\in \mathbb N\}$, or orbits $g^{\pm\infty}=\{g^i:i\in\mathbb Z\}$, respectively, of non-torsion elements~$g$ of the group $G$, where `sufficiently close' (forward) orbits become identified, together with a metric bounded by 1. We show that for all finitely generated groups, the distance between the antipodal points $g^\infty$ and $g^{-\infty}$ in the linear boundary is bounded from below by $\sqrt{1/2}$, and we give an example of a group which has two antipodal elements of distance at most $\sqrt{12/17}<1$. Our example is a derivation of the Baumslag-Gersten group. \newline We also exhibit a group with elements $g$ and $h$ such that $g^\infty = h^\infty$, but $g^{-\infty}\neq h^{-\infty}$. Furthermore, we introduce a notion of average-case-distortion---called growth---and compute explicit positive lower bounds for distances between points $g^\infty$ and $h^\infty$ which are limits of group elements $g$ and $h$ with different growth