 Hyperbolic group

In group theory, a hyperbolic group, also known as a word hyperbolic group, Gromov hyperbolic group, negatively curved group is a finitely generated group equipped with a word metric satisfying certain properties characteristic of hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by Mikhail Gromov in the early 1980s. He noticed that many results of Max Dehn concerning the fundamental group of a hyperbolic Riemann surface do not rely either on it having dimension two or even on being a manifold and hold in much more general context. In a very influential paper from 1987, Gromov proposed a wideranging research program. Ideas and foundational material in the theory of hyperbolic groups also stem from the work of George Mostow, William Thurston, James W. Cannon, Eliyahu Rips, and many others.
Contents
Definitions
Hyperbolic groups can be defined in several different ways. All definitions use the Cayley graph of the group and involve a choice of a positive constant δ and first define a δhyperbolic group. A group is called hyperbolic if it is δhyperbolic for some δ. When translating between different definitions of hyperbolicity, the particular value of δ may change, but the resulting notions of a hyperbolic group turn out to be equivalent.
Let G be a finitely generated group, and T be its Cayley graph with respect to some finite set S of generators. By identifying each edge isometrically with the unit interval in R, the Cayley graph becomes a metric space. The group G acts on T by isometries and this action is simply transitive on the vertices. A path in T of minimal length that connects points x and y is called a geodesic segment and is denoted [x,y]. A geodesic triangle in T consists of three points x, y, z, its vertices, and three geodesic segments [x,y], [y,z], [z,x], its sides.
The first approach to hyperbolicity is based on the slim triangles condition and is generally credited to Rips. Let δ > 0 be fixed. A geodesic triangle is δslim if each side is contained in a δneighborhood of the other two sides:
The Cayley graph T is δhyperbolic if all geodesic triangles are δslim, and in this case G is a δhyperbolic group. Although a different choice of a finite generating set will lead to a different Cayley graph and hence to a different condition for G to be δhyperbolic, it is known that the notion of hyperbolicity, for some value of δ is actually independent of the generating set. In the language of metric geometry, it is invariant under quasiisometries. Therefore, the property of being a hyperbolic group depends only on the group itself.
Remark
By imposing the slim triangles condition on geodesic metric spaces in general, one arrives at the more general notion of δhyperbolic space. Hyperbolic groups can be characterized as groups G which admit an isometric properly discontinuous action on a proper geodesic Δhyperbolic space X such that the factorspace X/G has finite diameter.
Examples of hyperbolic groups
 Finite groups.
 Virtually cyclic groups.
 Finitely generated free groups, and more generally, groups that act on a locally finite tree with finite stabilizers.
 Most surface groups are hyperbolic, namely, the fundamental groups of surfaces with negative Euler characteristic. For example, the fundamental group of the sphere with two handles (the surface of genus two) is a hyperbolic group.
 Most triangle groups Δ(l,m,n) are hyperbolic, namely, those for which 1/l + 1/m + 1/n < 1, such as the (2,3,7) triangle group.
 The fundamental groups of compact Riemannian manifolds with strictly negative sectional curvature.
 Groups that act cocompactly and properly discontinuously on a proper CAT(k) space with k < 0. This class of groups includes all the preceding ones as special cases. It also leads to many examples of hyperbolic groups not related to trees or manifolds.
 In some sense, "most" finitely presented groups with large defining relations are hyperbolic. See Random group.
Examples of nonhyperbolic groups
 The free rank 2 abelian group Z^{2} is not hyperbolic.
 More generally, any group which contains Z^{2} as a subgroup is not hyperbolic.^{[1]} In particular, lattices in higher rank semisimple Lie groups and the fundamental groups π_{1}(S^{3}−K) of nontrivial knot complements fall into this category and therefore are not hyperbolic.
 Baumslag–Solitar groups B(m,n) and any group that contains a subgroup isomorphic to some B(m,n) fail to be hyperbolic (since B(1,1) = Z^{2}, this generalizes the previous example).
 A nonuniform lattice in rank 1 semisimple Lie groups is hyperbolic if and only if the associated symmetric space is the hyperbolic plane.
Homological characterization
In 2002, I. Mineyev showed that hyperbolic groups are exactly those finitely generated groups for which the comparison map between the bounded cohomology and ordinary cohomology is surjective in all degrees, or equivalently, in degree 2.
Properties
Hyperbolic groups have a solvable word problem. They are biautomatic and automatic.^{[2]}: indeed, they are strongly geodesically automatic, that is, there is an automatic structure on the group, where the language accepted by the word acceptor is the set of all geodesic words.
In a 2010 paper^{[3]}, it was shown that hyperbolic groups have a decidable marked isomorphism problem. It is notable that this means that the isomorphism problem, orbit problems (in particular the conjugacy problem) and Whitehead's problem are all decidable.
Generalizations
Main article: Relatively hyperbolic groupAn important generalization of hyperbolic groups in geometric group theory is the notion of a relatively hyperbolic group. Motivating examples for this generalization are given by the fundamental groups of noncompact hyperbolic manifolds of finite volume, in particular, the fundamental groups of hyperbolic knots, which are not hyperbolic in the sense of Gromov.
A group G is relatively hyperbolic with respect to a subgroup H if, after contracting the Cayley graph of G along Hcosets, the resulting graph equipped with the usual graph metric is a δhyperbolic space and, moreover, it satisfies an additional technical condition which implies that quasigeodesics with common endpoints travel through approximately the same collection of cosets and enter and exit these cosets in approximately the same place.
Notes
 ^ Ghys and de la Harpe, Ch. 8, Th. 37; Bridson and Haefliger, Chapter 3.Γ, Corollary 3.10.
 ^ Charney, Ruth (1992), "Artin groups of finite type are biautomatic", Mathematische Annalen 292: 671–683, doi:10.1007/BF01444642
 ^ Dahmani, F.; Guirardel, V.  On the Isomorphism Problem in all Hyperbolic Groups, arXiV: 1002.2590
References
 Mikhail Gromov, Hyperbolic groups. Essays in group theory, 75263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987.
 Bridson, Martin R.; Haefliger, André (1999). Metric spaces of nonpositive curvature. Grundlehren der Mathematischen Wissenschaften 319. Berlin: SpringerVerlag. xxii+643. ISBN 3540643249. MR1744486
 Igor Mineyev, Bounded cohomology characterizes hyperbolic groups., Quart. J. Math. Oxford Ser., 53(2002), 5973.
Further reading
 É. Ghys and P. de la Harpe (editors), Sur les groupes hyperboliques d'après Mikhael Gromov. Progress in Mathematics, 83. Birkhäuser Boston, Inc., Boston, MA, 1990. xii+285 pp. ISBN 0817635084
 Michel Coornaert, Thomas Delzant and Athanase Papadopoulos, "Géométrie et théorie des groupes : les groupes hyperboliques de Gromov", Lecture Notes in Mathematics, vol. 1441, SpringerVerlag, Berlin, 1990, x+165 pp. MR 92f:57003, ISBN 3540529772
 Michel Coornaert and Athanase Papadopoulos, Symbolic dynamics and hyperbolic groups. Lecture Notes in Mathematics. 1539. SpringerVerlag, Berlin, 1993, viii+138 pp. ISBN 3540564993
Categories: Geometric group theory
 Metric geometry
 Properties of groups
 Combinatorics on words

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