Geodesic
Strong hyperbolicity
Bogdan Nica1 ; Ján Špakula2
1Burnside Hall, Montreal, Canada
2University of Southampton, UK
Groups, geometry, and dynamics, Tome 10 (2016) no. 3, pp. 951-964

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DOI

We propose the metric notion of strong hyperbolicity as a way of obtaining hyperbolicity with sharp additional properties. Specically, strongly hyperbolic spaces are Gromov hyperbolic spaces that are metrically well-behaved at infinity, and, under weak geodesic assumptions, they are strongly bolic as well. We show that CAT(–1) spaces are strongly hyperbolic. On the way, we determine the best constant of hyperbolicity for the standard hyperbolic plane H2. We also show that the Green metric defined by a random walk on a hyperbolic group is strongly hyperbolic. A measure-theoretic consequence at the boundary is that the harmonic measure defined by a random walk is a visual Hausdorff measure.
DOI : 10.4171/ggd/372
Classification : 20-XX
Mots-clés : Hyperbolic group, Green metric, CAT(–1) space, harmonic measure

Bogdan Nica  1   ; Ján Špakula  2

1 Burnside Hall, Montreal, Canada
2 University of Southampton, UK 
Bogdan Nica; Ján Špakula. Strong hyperbolicity. Groups, geometry, and dynamics, Tome 10 (2016) no. 3, pp. 951-964. doi: 10.4171/ggd/372
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     author = {Bogdan Nica and J\'an \v{S}pakula},
     title = {Strong hyperbolicity},
     journal = {Groups, geometry, and dynamics},
     pages = {951--964},
     year = {2016},
     volume = {10},
     number = {3},
     doi = {10.4171/ggd/372},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/372/}
}
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