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Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov. (Geometry and group theory. The hyperbolic groups of Gromov). (French) Zbl 0727.20018
Lecture Notes in Mathematics, 1441. Berlin etc.: Springer-Verlag. x, 165 p. DM 30.00 (1990).
These lecture notes contain an elaboration of M. Gromov’s article on hyperbolic groups [Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)]. To any group \(\Gamma\) with a finite set of generators G, we may assign the Cayley graph whose set of vertices is \(\Gamma\) and whose edges connect those \(\gamma,\gamma '\in \Gamma\) with \(\gamma '\gamma^{- 1}\in G\) (we may assume \(G=G^{-1})\). This becomes a length space (i.e. a metric space where the distances can be measured by arc length of curves) if we declare the length of any edge to be 1. Roughly speaking, a length space is called hyperbolic if each geodesic triangle is “\(\delta\)-thin”, i.e. (uniformly) quasi-isometric to a Y-shaped figure in the plane. E.g., complete simply connected Riemannian manifolds with curvature \(\leq -\delta\) are hyperbolic. Also, trees are hyperbolic. A finitely generated group \(\Gamma\) is called hyperbolic if its Cayley graph is hyperbolic. Since the notion of a hyperbolic length space is invariant under quasi-isometries, the choice of the generator set G does not matter. Geometric ideas, mainly from differential geometry of negative curvature, are used to derive algebraic properties of these groups.
Examples: 1. (Ch. 2-4) By an argument of Morse and Hedlund (1942), in a simply connected Riemannian manifold with curvature \(\leq -\delta\), quasi- geodesics (i.e. curves which satisfy locally \(length\leq a\cdot dis\tan ce+b\) where a, b are uniform constants) lie within a uniform distance from geodesics. This idea can be applied also to hyperbolic length spaces, in particular to a hyperbolic group \(\Gamma\), and it gives the solution of the word and the conjugacy problem in \(\Gamma\) : There is a constant L such that any trivial word \(1=g_ 1g_ 2...g_ N\) with \(g_ i\in G\) and \(N\geq L\) contains a subword of length \(\leq L\) which is not geodesic. Moreover, if two words w, \(w'\) (all of whose subwords of length L are geodesic) are conjugate, then \(w'=v\cdot \sigma (w)\cdot v^{-1}\) where v is a word of uniformly bounded length and \(\sigma\) denotes a cyclic permutation of the letters.
2. (Ch. 5) One can construct a contractible finite dimensional simplicial complex on which \(\Gamma\) acts freely with compact quotient. Hence, the cohomological dimension of \(\Gamma\) is finite.
3. (Ch. 7,8) The fundamental group of a Riemannian manifold in hyperbolic iff its universal covering manifold is a hyperbolic length space. Moreover, a simply connected Riemannian manifold is hyperbolic iff it satisfies a linear isoperimetric inequality: There is a constant k such that each closed curve C bounds a 2-disc D with area(D)\(\leq k\cdot length(C)\). This characterization can be used to show the hyperbolicity of certain amalgamated sums of the fundamental groups of two oriented manifolds \(M_ 1\), \(M_ 2\) with the same dimension \(\geq 4\), where \(M_ 1\) is compact with curvature \(\leq -\delta\) and \(M_ 2\) hyperbolic. One glues together \(M_ 1\) and \(M_ 2\) along a hypersurface \(\Sigma\) so that the new manifold M gets the new group as fundamental group. A suitable disk \(D\subset M\) is cutted by \(\Sigma\) into two pieces \(D_ j\subset M_ j\). One may assume that \(D_ 1\) is minimal and \(D\cap \Sigma\) geodesic. Using geometry of negative curvature in \(M_ 1\), one estimates the length of \(D\cap \Sigma\) from above so that the linear isoperimetric inequalities for \(D_ j\) give such an inequality for D.
4. (Ch. 8-10) Some of the convexity properties of negatively curved manifolds remain valid in hyperbolic length spaces. E.g., as a consequence it is proved that a hyperbolic group does not contain \({\mathbb{Z}}\oplus {\mathbb{Z}}\) as a subgroup.
Chapter 11 treats the boundary structure at infinity, in particular the action of isometries and quasi-isometries at infinity. In the last chapter (Ch. 12), the theory of automatic groups is developed, and it is shown that hyperbolic groups are automatic. As a consequence, the growth function \(f(t)=\sum c_ kt^ k\) where \(c_ k\) is the number of group elements of length k, is a rational function; one can derive a linear recursion formula for the coefficients \(c_ k\).

MSC:
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20-02 Research exposition (monographs, survey articles) pertaining to group theory
53C20 Global Riemannian geometry, including pinching
20F05 Generators, relations, and presentations of groups
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
57M05 Fundamental group, presentations, free differential calculus
57S25 Groups acting on specific manifolds
20E08 Groups acting on trees
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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