Uniformly Separated Sets and Gromov Hyperbolicity of Domains with the Quasihyperbolic Metric

Abstract

In this article we prove comparative results on the Gromov hyperbolicity of plane domains equipped with the quasihyperbolic metric. By a comparative result we mean one which assumes hyperbolicity in one domain and obtains it in a different domain somehow related to the original one. We derive a characterization (simple to check in practical cases) of the Gromov hyperbolicity of a plane domain Ω * obtained by deleting from the original domain Ω any uniformly separated union of compact sets. We present as well a result about stability of hyperbolicity.Mathematics Subject Classification (2010). Primary 30F45; Secondary 53C23, 30C99.

Gromov Hyperbolicity in Strong Product Graphs

“…For instance, the Gehring-Osgood j-metric is Gromov hyperbolic; and the Vuorinen j-metric is not Gromov hyperbolic except in the punctured space (see [21]). The study of Gromov hyperbolicity of the quasihyperbolic and the Poincaré metrics is the subject of [1,3,7,22,23,24,25,39,40,43,44,45,49]. In particular, the equivalence of the hyperbolicity of Riemannian manifolds and the hyperbolicity of a simple graph was proved in [39,43,45,49], hence, it is useful to know hyperbolicity criteria for graphs.…”

Gromov hyperbolicity of Denjoy domains through fundamental domains

“…However, as soon as simple connectedness is omitted, there is no immediate answer to whether the space Ω is hyperbolic or not. The question has lately been studied in [3], [18], [20], [19], [21], [27], [29]- [38] and [40].…”

“…The Gromov hyperbolicity of Denjoy domains with the Poincaré and quasihyperbolic metrics has been studied previously in [18], [20] and [21] in terms of the Euclidean size of the boundary of the Denjoy domain. The same topic, for the Poincaré metric only, has been dealt with in [3] and [33], but from a geometric point of view; the criteria so obtained involve the lengths of some kind of closed geodesics.…”