Δ-hyperbolic space | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-31

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Hyperbolic_metric_space


00:00:34 1 Definitions
00:01:17 1.1 Definition using the Gromov product
00:04:47 1.2 Definitions using triangles
00:10:06 1.3 Examples
00:11:00 1.4 Hyperbolicity and curvature
00:12:32 2 Important properties
00:14:36 2.1 Invariance under quasi-isometry
00:14:46 2.2 Approximate trees in hyperbolic spaces
00:16:08 2.3 Exponential growth of distance and isoperimetric inequalities
00:20:08 2.4 Quasiconvex subspaces
00:24:26 2.5 Asymptotic cones
00:25:31 3 The boundary of a hyperbolic space
00:25:50 3.1 Definition using the Gromov product
00:26:26 3.2 Definition for proper spaces using rays
00:30:43 3.3 Examples
00:32:37 3.4 Busemann functions
00:35:10 3.5 The action of isometries on the boundary and their classification
00:35:39 4 More examples
00:38:56 5 See also
00:40:00 6 Notes



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SUMMARY
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In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number δ) between points. The definition, introduced by Mikhael Gromov, generalizes the metric properties of classical hyperbolic geometry and of trees. Hyperbolicity is a large-scale property, and is very useful to the study of certain infinite groups called (Gromov-)hyperbolic groups.