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 Gromov (1987). 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.

Definitions

Hyperbolic groups can be defined in several different ways. Many 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.

$x$

$y$

$z$

$B_\delta([x,y])$

$B_\delta([z,x])$

$B_\delta([y,z])$

The δ-slim triangle condition

The first approach to hyperbolicity is based on the slim triangles condition of hyperbolic triangles and is generally credited to Rips. Let δ > 0 be fixed. A geodesic triangle is δ-slim if each side is contained in a $\delta$-neighborhood of the other two sides:

$[x,y] \subseteq B_{\delta}([y,z]\cup[z,x]),$

$[y,z]\subseteq B_{\delta}([z,x]\cup[x,y]),$

$[z,x]\subseteq B_{\delta}([x,y]\cup[y,z]).$

The Cayley graph T is δ-hyperbolic if all geodesic triangles are δ-slim, 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.

Remark

By imposing the slim 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.

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 $\Delta(l,m,n)$ 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 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. 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).
• A non-uniform 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. 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.

It was shown in 2010 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.

Cannon and Swenson have shown that hyperbolic groups with a 2-sphere at infinity have a natural subdivision rule. This is related to Cannon's conjecture.

Generalizations

Main article: Relatively hyperbolic group

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.