Geometric characterizations of virtually free groups

Abstract

Four geometric conditions on a geodesic metric space, which are stronger variants of classical conditions characterizing hyperbolicity (featuring -Thin polygons, the Gromov product or the mesh of triangles), are proved to be equivalent. They define the class of polygon hyperbolic geodesic metric spaces. In the particular case of the Cayley graph of a finitely generated group, it is shown that they characterize virtually free groups.