Quasi-geodesic segments and Gromov hyperbolic spaces
type: Item
instance of: scholarly article
title: Quasi-geodesic segments and Gromov hyperbolic spaces
zbMATH Open document ID: 0864.53031
DOI: 10.1007/BF00181569
published in: Geometriae Dedicata
publication date: 1997-06-17T00:00:00Z
review text: Let \(X\) be a geodesic metric space. The author makes the following definitions: For \(\lambda\geq 1\), a path \(\varphi: [a,b]\to X\) is said to be a \(\lambda\)-chord-arc curve if for every \([c,d] \subset[a,b]\), we have \[ \text{length} \varphi \bigl([c,d]\bigr) \leq \lambda \bigl|\varphi (c)- \varphi(d) \bigr|. \] The space \(X\) is said to be geodesically stable if for each \(\lambda\geq 1\), there exists \(M>0\) such that every \(\lambda\)-chord-arc curve \(\varphi: [a,b]\to X\) is contained in the \(M\)-neighborhood of some geodesic segment joining \(\varphi(a)\) and \(\varphi(b)\). Let \(t>0\). A path \(\varphi: [a,b]\to X\) is said to be a \(t\)-detour if there exists a point \(x\) on some geodesic segment joining \(\varphi(a)\) and \(\varphi (b)\) such that the image of \(\varphi\) does not intersect the closed ball centered at \(x\) of radius \(t\). The detour growth function of \(X\), \(G_X: ]0,\infty[\to[0,\infty]\) is defined for all \(t>0\) by \[ G_X(t)= \inf\bigl\{\text{length} (\varphi): \varphi \text{ is a } t-\text{detour} \bigr\}. \] Then the author proves that the following conditions are equivalent: (i) \(X\) is Gromov hyperbolic; (ii) \(X\) is geodesically stable (iii) \(\lim_{t\to\infty} G_X(t)/t=\infty\).
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 929919
zbMATH Keywords: hyperbolic space
zbMATH Keywords: geodesic stability
zbMATH Keywords: geodesic metric space
zbMATH Keywords: Gromov hyperbolic
author: Mario Bonk
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 929919