Elements of Asymptotic Geometry

Quasimöbius maps, weakly quasimöbius maps and uniform perfectness in quasi-metric spaces

“…The introduction of quasimöbius maps has provided a handy tool when studying the quasisymmetric maps and the quasiconformal maps. Many references related to the relationships among quasimöbius maps, quasisymmetric maps and quasiconformal maps have been in literature; see [1,5,6,7,13,15,16,17,18,19,20,21,25,29,30] etc. The precise definition for quasimöbius maps is as follows.…”

Metrics 𝜌, quasimetrics 𝜌^{𝑠} and pseudometrics inf𝜌^{𝑠}

“…Frink [9] used it for a proof of the metrizability of an f -quasimetric, which was simpler than in [7]. It is convenient to express some asymptotic properties of the metrics in the Gromov hyperbolic spaces ( [5,6,13]) in terms of the behavior of the function inf ρ s in its dependence on the parameter s. The main objective of this paper is to present some examples concerning the dependence of the geometry and topology of the space (X, inf ρ s ) on the index s. Our interest in this question has arisen in connection with the work [13], where a rather subtle example is constructed of a space X with degenerate metric inf ρ s , s > 1. In particular, in Basic Example 1, we present a simpler construction.…”

The fixed point property and unbounded sets in CAT(0) spaces

“…Still the space provided by Example 3.3. is δ-hyperconvex (see [4,Chapter 1] for definition and properties). Therefore one step farther in the above problem is to consider whether Ray's theorem fails on any non locally compact and complete CAT(0) space which is δ-hyperbolic for some δ ≥ 0.…”