Strongly geodesically automatic groups are hyperbolic
Due to the existence of a recursive structure on the set of all geodesics of a cocompact discrete group of hyperbolic isometries proved by \textit{J. Cannon} [Geom. Dedicata 16, 123-148 (1984; Zbl 0606.57003)] any hyperbolic group [in the sense of \textit{M. Gromov}, see Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)] is strongly geodesically automatic [see also \textit{D. Epstein} a.o., Word processing in groups (1992; Zbl 0764.20017)]. The main result of the paper is that the converse is also true: If a group is strongly geodesically automatic then it is hyperbolic. The proof of this fact is based on the author's observation that triangles in a graph are thin if bigons are thin. This also gives an alternative definition of hyperbolic groups, namely that a group is hyperbolic in the sense of Gromov if for some \(\delta>0\) bigons in its Cayley graph are \(\delta\)-thin.
cocompact discrete groups of hyperbolic isometries
scientific article; zbMATH DE number 807306