Strong hyperbolicity. Nica, B. & Spakula, J. *arXiv:1408.0250 [math]*, June, 2015. arXiv: 1408.0250Paper abstract bibtex We propose the metric notion of superbolicity as a way of obtaining hyperbolicity with sharp additional properties. Speciﬁcally, superbolic spaces are Gromov hyperbolic spaces that are as metrically well-behaved at inﬁnity as CAT(−1) spaces, and, under weak geodesic assumptions, they are strongly bolic as well. We show that CAT(−1) spaces are superbolic. On the way, we determine the best constant of hyperbolicity for H2. We also show that the Green metric deﬁned by a random walk on a hyperbolic group is superbolic. A measuretheoretic consequence at the boundary is that the harmonic measure deﬁned by a random walk is a visual Hausdorﬀ measure.

@article{nica_strong_2015,
title = {Strong hyperbolicity},
url = {http://arxiv.org/abs/1408.0250},
abstract = {We propose the metric notion of superbolicity as a way of obtaining hyperbolicity with sharp additional properties. Speciﬁcally, superbolic spaces are Gromov hyperbolic spaces that are as metrically well-behaved at inﬁnity as CAT(−1) spaces, and, under weak geodesic assumptions, they are strongly bolic as well. We show that CAT(−1) spaces are superbolic. On the way, we determine the best constant of hyperbolicity for H2. We also show that the Green metric deﬁned by a random walk on a hyperbolic group is superbolic. A measuretheoretic consequence at the boundary is that the harmonic measure deﬁned by a random walk is a visual Hausdorﬀ measure.},
language = {en},
urldate = {2020-12-14},
journal = {arXiv:1408.0250 [math]},
author = {Nica, Bogdan and Spakula, Jan},
month = jun,
year = {2015},
note = {arXiv: 1408.0250},
keywords = {20F67, Mathematics - Group Theory, Mathematics - Metric Geometry},
}

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