Linear and projective boundaries in HNN-extensions and distortion phenomena

TL;DR

This paper studies the geometric boundaries of finitely generated groups, providing bounds on distances between boundary points, and introduces a new concept called growth to analyze distortion phenomena.

Contribution

It establishes bounds on antipodal points in linear boundaries, provides examples with unusual boundary behaviors, and introduces the notion of growth for analyzing distortion.

Findings

Lower bound of .707 for antipodal points in linear boundary

Example of a group with boundary points at distance .707

Introduction of growth as a measure of distortion with explicit bounds

Abstract

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.