Linear and projective boundaries in HNN-extensions and distortion phenomena

Abstract

Linear and projective boundaries of Cayley graphs were introduced in [6] as quasi-isometry invariant boundaries of finitely generated groups. They consist of forward orbits g1 D ¹gi W i 2 Nº, or orbits g 1 D ¹gi W i 2 Zº, 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 g1 and gð€€€1 in the linear boundary is bounded from below by p 1=2, and we give an example of a group which has two antipodal elements of distance at most p 12=17 < 1. Our example is a derivation of the Baumslag–Gersten group. We also exhibit a group with elements g and h such that g1 D h1, but gð€€€1 ¤ hð€€€1. Furthermore, we introduce a notion of average-case-distortion—called growth—and compute explicit positive lower bounds for distances between points g1 and h1 which are limits of group elements g and h with different growth.