Gromov hyperbolicity of Denjoy domains with hyperbolic and quasihyperbolic metrics

TL;DR

This paper provides explicit criteria based on Euclidean size to determine Gromov hyperbolicity of Denjoy domains with Poincaré or quasihyperbolic metrics, including conditions for non-hyperbolicity of periodic domains.

Contribution

It introduces simple, explicit conditions for Gromov hyperbolicity of Denjoy domains, linking geometric properties to hyperbolic metric behavior, and applies these to periodic domains.

Findings

Criteria based on Euclidean size determine hyperbolicity

Periodic Denjoy domains are generally non-hyperbolic

Conditions are explicit and easy to verify

Abstract

We obtain explicit and simple conditions which in many cases allow one decide, whether or not a Denjoy domain endowed with the Poincare or quasihyperbolic metric is Gromov hyperbolic. The criteria are based on the Euclidean size of the complement. As a corollary, the main theorem allows to deduce the non-hyperbolicity of any periodic Denjoy domain.