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MiS Preprint
62/2012
Linear and Projective Boundaries in HNN-Extensions and Distortion Phenomena
Bernhard Krön, Jörg Lehnert and Maya Stein
Abstract
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 $g^\infty=\{g^i: i\in \mathbb N\}$ 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 $g^{\pm\infty}=\{g^i:i\in\mathbb Z\}$ 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 $g^\infty$ and $g^{-\infty}$ 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 $g^\infty$ and $g^{-\infty}$ is less or equal $(12/17)^{1/2}$.
We also give an example of a group with elements $g$ and $h$ such that $g^\infty = h^\infty$, but $g^{-\infty}\neq h^{-\infty}$. 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 $g^\infty$ and $h^\infty$ which are limits of group elements $g$ and $h$ with different growth.
