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 $g^{\mathrm{\infty }}=\{g^i:i\in \mathbb{N}\}$, or orbits $g^{\pm \mathrm{\infty }}=\{g^i:i\in \mathbb{Z}\}$, 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 $g^{\mathrm{\infty }}$ and $g^{-\mathrm{\infty }}$ in the linear boundary is bounded from below by $\sqrt{1\mathrm{/}2}$, and we give an example of a group which has two antipodal elements of distance at most $\sqrt{12\mathrm{/}17}<1$. Our example is a derivation of the Baumslag-Gersten group. \newline We also exhibit a group with elements $g$ and $h$ such that $g^{\mathrm{\infty }}=h^{\mathrm{\infty }}$, 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 $g^{\mathrm{\infty }}$ and $h^{\mathrm{\infty }}$ which are limits of group elements $g$ and $h$ with different growth.