Hyperbolic group

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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 wide-ranging 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 1 Examples of hyperbolic groups 2 Examples of non-hyperbolic groups 3 Definitions 3.1 Remark 4 Properties 5 Generalizations 6 Notes 7 References 8 Further reading

[edit] 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 discontinuosly 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.

[edit] Examples of non-hyperbolic groups

• The free rank 2 abelian group Z² is not hyperbolic.

• More generally, any group which contains Z² as a subgroup is not hyperbolic.^[1] In particular, lattices in higher rank semisimple Lie groups and the fundamental groups π₁(S³−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) = Z², this generalizes the previous example).

• Non-uniform lattices in rank 1 semisimple Lie groups are not hyperbolic.

[edit] 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 δ-hyperbolic group. A group is called hyperbolic if it is δ-hyperbolic 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 thin triangles condition and is generally credited to Rips. Let be fixed. A geodesic triangle is δ-thin if each side is contained in a δ-neighborhood of the other two sides:

The Cayley graph T is δ-hyperbolic if all geodesic triangles are δ-thin, and in this case G is a δ-hyperbolic 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 δ-hyperbolic, 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 quasi-isometries. Therefore, the property of being a hyperbolic group depends only on the group itself.

[edit] Remark

By imposing the thin triangles condition on geodesic metric spaces in general, one arrives at the more general notion of δ-hyperbolic space. Hyperbolic groups can be characterized as groups G which admit an isometric properly discontinuous action on a proper geodesic Δ-hyperbolic space X such that the factor-space X/G has finite diameter.

[edit] Properties

Hyperbolic groups have a soluble word problem. Indeed, they are strongly geodesically automatic. That is, there is an automatic structure of the group, where the language accepted by the word acceptor is the set of all geodesic words.

[edit] Generalizations

An 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 non-compact 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 H-cosets, the resulting graph equipped with the usual graph metric is a δ-hyperbolic space and, moreover, it satisfies an additional technical condition which implies that quasi-geodesics with common endpoints travel through approximately the same collection of cosets and enter and exit these cosets in approximately the same place.

[edit] Notes

1. ^ Ghys and de la Harpe, Ch. 8, Th. 37; Bridson and Haefliger, Chapter 3.Γ, Corollary 3.10.

[edit] References

• Mikhail Gromov, Hyperbolic groups. Essays in group theory, 75--263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987.

• Bridson, Martin R.; Haefliger, André (1999). Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften 319. Berlin: Springer-Verlag. xxii+643. ISBN 3-540-64324-9.  MR1744486

[edit] 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 0-8176-3508-4
• Michel Coornaert, Thomas Delzant, Athanase Papadopoulos, "Géométrie et théorie des groupes : les groupes hyperboliques de Gromov", Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990, MR 92f:57003, ISBN 3-540-52977-2