Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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QUASI-ISOMETRIC AND WEAKLY QUASISYMMETRIC MAPS BETWEEN LOCALLY COMPACT NON-COMPLETE METRIC SPACES

  • Received : 2017.05.11
  • Accepted : 2017.09.14
  • Published : 2018.05.31
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Abstract

The aim of this paper is to show that there exists a weakly quasisymmetric homeomorphism $f:(X,d){\rightarrow}(Y,d^{\prime})$ between two locally compact non-complete metric spaces such that $f:(X,d_h){\rightarrow}(Y,d^{\prime}_h)$ is not quasi-isometric, where dh denotes the Gromov hyperbolic metric with respect to the metric d introduced by Ibragimov in 2011. This result shows that the answer to the related question asked by Ibragimov in 2013 is negative.

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References

  1. Z. Ibragimov, Hyperbolizing metric spaces, Proc. Amer. Math. Soc. 139 (2011), no. 12, 4401-4407. https://doi.org/10.1090/S0002-9939-2011-10857-8
  2. Z. Ibragimov, Hyperbolization of locally compact non-complete metric spaces, in In the tradition of Ahlfors-Bers. VI, 51-59, Contemp. Math., 590, Amer.Math. Soc., Providence, RI, 2013.