Geodesic
Rough isometry between Gromov hyperbolic spaces and uniformization
Jeff Lindquist1 ; Nageswari Shanmugalingam1
1 University of Cincinnati, Department of Mathematical Sciences
Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 449-464

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  In this note we show that given two complete geodesic Gromov hyperbolic spaces that are roughly isometric and an arbitrary $\epsilon>0$ (not necessarily small), either the uniformization of both spaces with parameter $\epsilon$ results in uniform domains, or else neither uniformized space is a uniform domain. The terminology of "uniformization" is from [BHK], where it is shown that the uniformization, with parameter $\epsilon>0$, of a complete geodesic Gromov hyperbolic space results in a uniform domain provided $\epsilon$ is small enough.
Keywords: Gromov hyperbolic, uniform domain, rough isometry, uniformization

Jeff Lindquist 1 ; Nageswari Shanmugalingam 1

1 University of Cincinnati, Department of Mathematical Sciences 
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Jeff Lindquist; Nageswari Shanmugalingam. Rough isometry between Gromov hyperbolic spaces and uniformization. Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 449-464. http://geodesic.mathdoc.fr/item/AFM_2021_46_1_a25/