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 \(c_ k\).

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 \(c_ k\).

Reviewer: J.-H.Eschenburg (Augsburg)

##### 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 |