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The theory of negatively curved spaces and groups. (English) Zbl 0764.57002
Ergodic theory, symbolic dynamics, and hyperbolic spaces, Lect. Workshop Hyperbolic Geom. Ergodic Theory, Trieste/Italy 1989, 315-369 (1991).
[For the entire collection see Zbl 0743.00040.]
The main motivation of this remarkable survey is that essentially all geometric constructs which are global in nature, such as paths of shortest length, global manifestations of curvature, planes, half-spaces, rates of growth, which are studied in differential geometry have manifestations in combinatorial approximation to that geometry. A good example of that is the characterization theorem of J. W. Cannon and D. Cooper [Trans. Am. Math. Soc. 330, No. 1, 419-431 (1992; Zbl 0761.57008)]: A group \(G\) acts geometrically on the hyperbolic 3-space \(H^ 3\) iff its Cayley graph is quasi-isometric with \(H^ 3\).
This survey contains 3 sections. First one discusses geometric group theory and negatively curved groups as presented by J. W. Cannon [Geom. Dedicata 16, 123-148 (1984; Zbl 0606.57003)] and M. Gromov [Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)]. In the second section, the author describes three solutions to the word problem in negatively curved groups: a generalized Dehn algorithm, a solution by finite state automata (automatic groups) and a solution by generalized cellular automata (almost convex groups). The last section discusses the problem of constant negative curvature, in particular a theorem characterizing finitely generated groups \(G\) which can act geometrically on \(H^ 3\). The necessary and sufficient conditions of it are: \(G\) is negatively curved; the space at infinity is a topological 2-sphere; and a condition considering some sets at infinity describable completely in terms of the combinatorics of the group \(G\). This is related to W. P. Thurston’s “Geometrization Conjecture” [Bull. Am. Math. Soc., New Ser. 6, 357-379 (1982; Zbl 0496.57005)] and Combinatorial Riemann Mapping Theorem (the author gives an outline of the proof of this theorem). In the Appendix, the author describes geometric actions of a group on an “\(n\)-connected geometry” related to the combinatorics of the group.

57M07 Topological methods in group theory
57S30 Discontinuous groups of transformations
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
53C99 Global differential geometry
57M05 Fundamental group, presentations, free differential calculus
57M50 General geometric structures on low-dimensional manifolds