Quasi-geodesic segments and Gromov hyperbolic spaces
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) \(\lim_{t\to\infty} G_X(t)/t=\infty\).
scientific article; zbMATH DE number 929919