Comparative Gromov hyperbolicity results for the hyperbolic and quasihyperbolic metrics

Hästö, Peter; Portilla, Ana; Rodríguez, José M.; Tourís, Eva

doi:10.1080/17476930902999033

Cited by 9 publications

(4 citation statements)

References 22 publications

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“…Whether or not this is true in general domains remains an open question. This paper is a continuation of [13], by the authors and Lindén, and we continue the work in [14]. The main innovation compared to that study is that we now work with quasigeodesics instead of geodesics.…”

Section: Introductionmentioning

confidence: 81%

Gromov hyperbolic equivalence of the hyperbolic and quasihyperbolic metrics in Denjoy domains

Hästö

Portilla²,

Rodríguez³

et al. 2010

Bulletin of the London Mathematical Society

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In this article we investigate the Gromov hyperbolicity of Denjoy domains equipped with the hyperbolic or the quasihyperbolic metric. We first prove the existence of suitable families of quasigeodesics. The main result shows that a Denjoy domain is Gromov hyperbolic with respect to the hyperbolic metric if and only it is Gromov hyperbolic with respect to the quasihyperbolic metric. Using these tools we give a characterization in terms of Euclidean distances of when the domains are Gromov hyperbolic. We also give several concrete examples of families of domains satisfying the criteria of the theorems.

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Section: Introductionmentioning

confidence: 81%

Gromov hyperbolic equivalence of the hyperbolic and quasihyperbolic metrics in Denjoy domains

Hästö

Portilla²,

Rodríguez³

et al. 2010

Bulletin of the London Mathematical Society

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Gromov Hyperbolicity in Strong Product Graphs

Carballosa

Casablanca

Cruz

et al. 2013

Electron. J. Combin.

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If X is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\delta$-hyperbolic $($in the Gromov sense$)$ if any side of $T$ is contained in a $\delta$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. If $X$ is hyperbolic, we denote by $\delta (X)$ the sharp hyperbolicity constant of $X$, i.e., $\delta (X)=\inf\{\delta\geq 0: \, X \, \text{ is $\delta$-hyperbolic}\,\}\,.$ In this paper we characterize the strong product of two graphs $G_1\boxtimes G_2$ which are hyperbolic, in terms of $G_1$ and $G_2$: the strong product graph $G_1\boxtimes G_2$ is hyperbolic if and only if one of the factors is hyperbolic and the other one is bounded. We also prove some sharp relations between $\delta (G_1\boxtimes G_2)$, $\delta (G_1)$, $\delta (G_2)$ and the diameters of $G_1$ and $G_2$ (and we find families of graphs for which the inequalities are attained). Furthermore, we obtain the exact values of the hyperbolicity constant for many strong product graphs.

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“…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].…”

Section: Introductionmentioning

confidence: 99%

Gromov hyperbolicity of Denjoy domains through fundamental domains

Hästö¹,

Portilla²,

Rodrı́guez³

et al. 2012

Publ. Math. Debrecen

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In this article we investigate the Gromov hyperbolicity of Denjoy domains equipped with their Poincaré metrics. This point of view allows us to improve some known results. We give a characterization in terms of fundamental domains of when the domains are Gromov hyperbolic. We also obtain criteria (easily applicable in practical cases) which allow us to decide when a Denjoy domain is either hyperbolic or not. Furthermore, for a special kind of domains, we can characterize its hyperbolicity in a very simple way.

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Comparative Gromov hyperbolicity results for the hyperbolic and quasihyperbolic metrics

Cited by 9 publications

References 22 publications

Gromov hyperbolic equivalence of the hyperbolic and quasihyperbolic metrics in Denjoy domains

Gromov hyperbolic equivalence of the hyperbolic and quasihyperbolic metrics in Denjoy domains

Gromov Hyperbolicity in Strong Product Graphs

Gromov hyperbolicity of Denjoy domains through fundamental domains

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