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Hyperbolic groups. (English) Zbl 0634.20015
Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987).
[For the entire collection see Zbl 0626.00014.]
The author gives three equivalent abstract definitions of (word) hyperbolic or negatively-curved groups. These groups include, but are not restricted to, the fundamental groups of closed negatively curved manifolds. The author shows how the curvature manifests itself in the combinatorics of the group. The author generalizes small-cancellation theory to this setting. The author exhibits numerous constructions which, when applied to hyperbolic groups, yield hyperbolic groups. The methods are general, powerful, and beautiful. The reader needs to beware that many of the proofs and statements, while correct in spirit, are incorrect in detail.
Reviewer: J.W.Cannon

MSC:
20F65 Geometric group theory
20F06 Cancellation theory of groups; application of van Kampen diagrams
57M05 Fundamental group, presentations, free differential calculus
20F05 Generators, relations, and presentations of groups
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
53A35 Non-Euclidean differential geometry
53C22 Geodesics in global differential geometry