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.

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.

Reviewer: B.N.Apanasov (Norman)

##### MSC:

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 |

##### Keywords:

automatic groups; almost convex groups; Geometrization Conjecture; hyperbolic 3-space; Cayley graph; negatively curved groups; word problem; generalized Dehn algorithm; finite state automata; generalized cellular automata; space at infinity; Combinatorial Riemann Mapping Theorem
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\textit{J. W. Cannon}, in: Ergodic theory, symbolic dynamics, and hyperbolic spaces. Lectures given at the workshop ''Hyperbolic geometry and ergodic theory'', held at the International Centre for Theoretical Physics in Trieste, Italy, 17-28 April, 1989. Oxford etc.: Oxford University Press. 315--369 (1991; Zbl 0764.57002)