MiS Preprint Repository

The linear boundary and the projective boundary have recently been introduced by Krön, Lehnert, Seifter and Teufl as a quasi-isometry invariant boundary of Cayley graphs of finitely generated groups, but also as a more general concept in metric spaces.

An element of the linear boundary of a Cayley graph is an equivalence class of forward orbits $g^{\mathrm{\infty }}=\{g^i:i\in \mathbb{N}\}$ of non-torsion elements $g$ of the group $G$. Two orbits are equivalent when they stay sublinearly close to each other. For a formal definition see below. The elements of the projective boundary are obtained by taking cyclic subgroups $g^{\pm \mathrm{\infty }}=\{g^i:i\in \mathbb{Z}\}$ instead of forward orbits. The boundaries are then obtained by equipping these points at infinity with an angle metric. A typical example is the $(n-1)$-dimensional sphere as linear boundary of ${\mathbb{Z}}^n$. Its projective boundary is the $(n-1)$-dimensional projective space.

The diameter of these boundaries is always at most 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 $(1/2{)}^{1/2}$. But these distances can actually be smaller than 1: we give an example of a one-relator group - a derivation of the Baumslag-Gersten group - which has an infinitely iterated HNN-extension as an isometrically embedded subgroup. In this example, there is an element $g$ for which the distance between $g^{\mathrm{\infty }}$ and $g^{-\mathrm{\infty }}$ is less or equal $(12/17{)}^{1/2}$.

We also give an example of a group with elements $g$ and $h$ such that $g^{\mathrm{\infty }}=h^{\mathrm{\infty }}$, but $g^{-\mathrm{\infty }}\ne h^{-\mathrm{\infty }}$. Furthermore, we introduce a notion of average-case-distortion -- called growth of elements -- and compute an explicit positive lower bound for the distances between points $g^{\mathrm{\infty }}$ and $h^{\mathrm{\infty }}$ which are limits of group elements $g$ and $h$ with different growth.