Strongly geodesically automatic groups are hyperbolic
type: Item
instance of: scholarly article
title: Strongly geodesically automatic groups are hyperbolic
zbMATH Open document ID: 0834.20040
DOI: 10.1007/BF01884301
published in: Inventiones Mathematicae
publication date: 1996-03-18T00:00:00Z
full work available at URL: https://eudml.org/doc/144301
review text: 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.
reviewed by: Boris N. Apanasov
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 57S30
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 807306
zbMATH Keywords: recursive structures
zbMATH Keywords: geodesics
zbMATH Keywords: cocompact discrete groups of hyperbolic isometries
zbMATH Keywords: strongly geodesically automatic groups
zbMATH Keywords: hyperbolic groups
zbMATH Keywords: Cayley graphs
author: Panos Papasoglu
MaRDI profile type: Publication
cites work: The combinatorial structure of cocompact discrete hyperbolic groups
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description: scientific article; zbMATH DE number 807306