Gromov hyperbolicity of Denjoy Domains

Álvarez, Venancio; Portilla, Ana; Rodríguez, José M.; Tourís, Eva

doi:10.1007/s10711-006-9102-z

Cited by 21 publications

(24 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…1 On the other hand, the Poincaré half space in R k with the hyperbolic metric is δ-hyperbolic with δ = log 2 3. Several classes of geodesic metric spaces are known to be hyperbolic [6,42] …”

Section: Introductionmentioning

confidence: 99%

Additive Spanners and Distance and Routing Labeling Schemes for Hyperbolic Graphs

et al. 2010

View full text Add to dashboard Cite

International audience$\delta$-Hyperbolic metric spaces have been defined by M. Gromov in 1987 via a simple 4-point condition: for any four points $u,v,w,x$, the two larger of the distance sums $d(u,v)+d(w,x), d(u,w)+d(v,x), d(u,x)+d(v,w)$ differ by at most $2\delta$. They play an important role in geometric group theory, geometry of negatively curved spaces, and have recently become of interest in several domains of computer science, including algorithms and networking. For example, (a) it has been shown empirically that the internet topology embeds with better accuracy into a hyperbolic space than into an Euclidean space of comparable dimension, (b) every connected finite graph has an embedding in the hyperbolic plane so that the greedy routing based on the virtual coordinates obtained from this embedding is guaranteed to work. A connected graph $G=(V,E)$ equipped with standard graph metric $d_G$ is $\delta$-{\it hyperbolic} if the metric space $(V,d_G)$ is $\delta$-hyperbolic. In this paper, using our Layering Partition technique, we provide a simpler construction of distance approximating trees of $\delta$-hyperbolic graphs on $n$ vertices with an additive error $O(\delta\log n)$ and show that every $n$-vertex $\delta$-hyperbolic graph has an additive $O(\delta \log n)$-spanner with at most $O(\delta n)$ edges. As a consequence, we show that the family of $\delta$-hyperbolic graphs with $n$ vertices enjoys an $O(\delta\log n)$-additive routing labeling scheme with $O(\delta\log^2n)$ bit labels and $O(\log\delta)$ time routing protocol, and an easier constructable $O(\delta\log n)$-additive distance labeling scheme with $O(\log^2n)$ bit labels and constant time distance decoder

show abstract

Section: Introductionmentioning

confidence: 99%

Additive Spanners and Distance and Routing Labeling Schemes for Hyperbolic Graphs

et al. 2010

View full text Add to dashboard Cite

show abstract

“…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], [22]- [26], [34]- [45] and [47].…”

Section: Introductionmentioning

confidence: 99%

“…The same topic, just for the Poincaré metric, has been dealt with in [3] and [38], but from a geometric point of view.…”

Section: Introductionmentioning

confidence: 99%

A Very Simple Characterization of Gromov Hyperbolicity for a Special Kind of Denjoy Domains

Portilla¹,

Rodrı́guez²,

Tourís³

2011

Journal of the Korean Mathematical Society

Self Cite

View full text Add to dashboard Cite

Abstract. In this paper we provide characterizations for the Gromov hyperbolicity of some particular Denjoy domains and besides some sufficient conditions to guarantee or discard the hyperbolicity of some others. The conditions obtained involve just the lengths of some special simple closed geodesics in the domain. These results, on the one hand, show the extraordinary complexity of determining the hyperbolicity of a domain and, on the other hand, allow us to construct easily a large variety of both hyperbolic and non-hyperbolic domains.

show abstract

“…However, as soon as simply connectedness is omitted, there is no immediate answer to whether the space h Ω is hyperbolic or not. The question has lately been studied in [3], [13], [16]- [18] and [23]- [32].…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2010

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Gromov hyperbolicity of Denjoy Domains

Cited by 21 publications

References 28 publications

Additive Spanners and Distance and Routing Labeling Schemes for Hyperbolic Graphs

Additive Spanners and Distance and Routing Labeling Schemes for Hyperbolic Graphs

A Very Simple Characterization of Gromov Hyperbolicity for a Special Kind of Denjoy Domains

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

Contact Info

Product

Resources

About