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

Papasoglu, P.

Invent. Math. 121, No. 2, 323-334 (1995).

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

Citations:

Zbl 0606.57003; Zbl 0634.20015; Zbl 0764.20017

References:

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[2]	[C] J.W. Cannon: The Combinatorial structure of cocompact discrete hyperbolic groups. Geometriae Dedicata16 (1984) 123–148 · Zbl 0606.57003
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