Abstract
A real valued function \(\varphi \) of one variable is called a metric transform if for every metric space (X, d) the composition \(d_\varphi = \varphi \circ d\) is also a metric on X. We give a complete characterization of the class of approximately nondecreasing, unbounded metric transforms \(\varphi \) such that the transformed Euclidean half line \(([0,\infty ),|\cdot |_\varphi )\) is Gromov hyperbolic. As a consequence, we obtain metric transform rigidity for roughly geodesic Gromov hyperbolic spaces, that is, if (X, d) is any metric space containing a rough geodesic ray and \(\varphi \) is an approximately nondecreasing, unbounded metric transform such that the transformed space \((X,d_\varphi )\) is Gromov hyperbolic and roughly geodesic then \(\varphi \) is an approximate dilation and the original space (X, d) is Gromov hyperbolic and roughly geodesic.
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The second author was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. The authors would also like to thank the referee for many valuable comments.
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Dragomir, G., Nicas, A. Metric transforms yielding Gromov hyperbolic spaces. Geom Dedicata 200, 331–350 (2019). https://doi.org/10.1007/s10711-018-0374-x
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DOI: https://doi.org/10.1007/s10711-018-0374-x