Linear and projective boundaries in HNN-extensions and distortion phenomena

Abstract

Abstract Linear and projective boundaries of Cayley graphs were introduced in [Glasg. Math. J. (2014), DOI 10.1017/S0017089514000512] as quasi-isometry invariant boundaries of finitely generated groups. They consist of forward orbits g ∞ = {gi : i ∈ ℕ}, or orbits g ±∞ = {gi : i ∈ 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 ∞ and g -∞ in the linear boundary is bounded from below by 1/21/2, and we give an example of a group which has two antipodal elements of distance at most (12/17)1/2 < 1. Our example is a derivation of the Baumslag–Gersten group. We also exhibit a group with elements g and h such that g ∞ = h ∞, but g -∞ ≠ h -∞. Furthermore, we introduce a notion of average-case-distortion—called growth—and compute explicit positive lower bounds for distances between points g ∞ and h ∞ which are limits of group elements g and h with different growth.