Strongly geodesically automatic groups are hyperbolic. (English) Zbl 0834.20040

Due to the existence of a recursive structure on the set of all geodesics of a cocompact discrete group of hyperbolic isometries proved by J. Cannon [Geom. Dedicata 16, 123-148 (1984; Zbl 0606.57003)] any hyperbolic group [in the sense of M. Gromov, see Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)] is strongly geodesically automatic [see also 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.


20F65 Geometric group theory
20F05 Generators, relations, and presentations of groups
57M05 Fundamental group, presentations, free differential calculus
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
57S30 Discontinuous groups of transformations
53C22 Geodesics in global differential geometry
Full Text: DOI EuDML


[1] [ABC] Alonso, Brady, Cooper, Delzant, Ferlini, Lustig, Mihalik, Shapiro, Short: Group Theory from a Geometrical Viewpoint, E. Ghys, A. Haefliger, A. Verjovsky, eds, World Scientific, 1991
[2] [C] J.W. Cannon: The Combinatorial structure of cocompact discrete hyperbolic groups. Geometriae Dedicata16 (1984) 123–148 · Zbl 0606.57003
[3] [ECHLPT] D. Epstein, J. Cannon, D. Holt, S. Levy, M. Paterson, W. Thurston: Word Processing in Groups, Jones and Bartlett, 1992
[4] [G] M. Gromov: Hyperbolic Groups. In: Essays in Group Theory, S.M. Gersten (ed.) MSRI series 8, Springer: Berlin Heidelberg New York, pp. 75–263
[5] [P] P. Papasoglu: On the subquadratic isoperimetric inequality. Submitted for publication to the Proceedings of the Ohio Geometric Group Theory Conference, 1992
[6] [Po] J. Pomroy, M.Sci. thesis, in preparation. Warwick University
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.