Gromov hyperbolicity through decomposition of metric spaces

Abstract

In this article we study the hyperbolicity in the Gromov sense of metric spaces. We deduce the hyperbolicity of a space from the hyperbolicity of its “building block components,” which can be joined following an arbitrary scheme. These results are especially valuable since they simplify notably the topology and allow to obtain global results from local information. Some interesting theorems about the role of punctures and funnels on the hyperbolicity of Riemann surfaces can be deduced from the conclusions of this article.