Linear and Projective Boundaries in HNN-Extensions and Distortion Phenomena

Bernhard Krön, Jörg Lehnert, and Maya Stein

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Submission date: 08. Oct. 2012
Pages: 22
published in: Journal of group theory, 18 (2015) 3, p. 455-488 
DOI number (of the published article): 10.1515/jgth-2015-0002
Bibtex
MSC-Numbers: 20F65, 20E06, 05C63
Keywords and phrases: HNN-extensions, boundaries, Baumslag-Gersten group, subgroup distortion, growth
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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^∞ = {gⁱ : i ∈ 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^±∞ = {gⁱ : i ∈ 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 ℤⁿ. 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^∞ and g^-∞ 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^∞ and g^-∞ is less or equal (12∕17)^1∕2.

We also give an example of 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 of elements – and compute an explicit positive lower bound for the distances between points g^∞ and h^∞ which are limits of group elements g and h with different growth.