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**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.

Reviewer: B.N.Apanasov (Norman)

### MSC:

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 |

### Keywords:

recursive structures; geodesics; cocompact discrete groups of hyperbolic isometries; strongly geodesically automatic groups; hyperbolic groups; Cayley graphs
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\textit{P. Papasoglu}, Invent. Math. 121, No. 2, 323--334 (1995; Zbl 0834.20040)

### References:

[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 |

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