A real variable characterization of Gromov hyperbolicity of flute surfaces

Abstract

In this paper we give a characterization of the Gromov hyperbolicity of trains (a large class of Denjoy domains which contains the flute surfaces) in terms of the behavior of a real function. This function describes somehow the distances between some remarkable geodesics in the train. This theorem has several consequences; in particular, it allows to deduce a result about stability of hyperbolicity, even though the original surface and the modified one are not quasi-isometric. In order to obtain these results we also prove some trigonometric lemmas that are interesting by themselves, since they provide very simple estimates on some hyperbolic distances.