Hyperbolic distance versus quasihyperbolic distance in plane domains

Herron, David; Lindquist, Jeff

doi:10.1090/btran/73

Hyperbolic distance versus quasihyperbolic distance in plane domains
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by David A. Herron and Jeff Lindquist HTML | PDF

Trans. Amer. Math. Soc. Ser. B 8 (2021), 578-614

Abstract:

We examine Euclidean plane domains with their hyperbolic or quasihyperbolic distance. We prove that the associated metric spaces are quasisymmetrically equivalent if and only if they are bi-Lipschitz equivalent. On the other hand, for Gromov hyperbolic domains, the two corresponding Gromov boundaries are always quasisymmetrically equivalent. Surprisingly, for any finitely connected hyperbolic domain, these two metric spaces are always quasiisometrically equivalent. We construct examples where the spaces are not quasiisometrically equivalent.

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