Hyperbolic group

Hyperbolic groups (also: word - hyperbolic groups, Gromov - hyperbolic groups, negatively curved groups) are one of the central themes of geometric group theory.

The term was introduced in the 1980s by Mikhail Leonidovich Gromov, the use of geometric methods in group theory but has a long hyperbolic up to Max Dehn use geometry to solve the word problem for fundamental groups of compact surfaces standing tradition. In a sense, almost all groups are hyperbolic. Numerous methods from the geometry of negatively curved spaces can be applied to hyperbolic groups and thus make it usable for the group theory.

Definition

A finitely generated group is hyperbolic if a finite system of generators associated Cayley graph is δ - hyperbolic for δ > 0. This definition is independent of the choice of the finite generating system.

For more details:

The assigned to a finite generating set S of a group G Cayley graph is defined as follows graph (V, E ): The node set V is the group G, the edge set E consists of pairs of the form ( g, gs ), where g is a any Grupp Enel element and s is an item. The picture on the right shows the Cayley graph of S of two elements = {a, b} generated free group.

By specifying that all edges have length 1, the Cayley graph becomes a metric space. ( The induced metric on the vertex set G is called the word metric of the group G. )

For several finite systems of generators is obtained quasi- isometric Cayley. All given up to quasi- isometry geometric properties of graphene thus correspond to properties of groups.

A metric space is called δ - hyperbolic for a δ > 0 if all geodesic triangles are δ - thin, ie each edge of the triangle is contained in the δ - neighborhood of the union of the other two edges:

This condition is met with, for example, for geodesic triangles in trees or in the hyperbolic plane, generally for geodesic triangles in simply connected Riemannian manifolds of negative sectional curvature.

If two metric spaces are quasi- isometric and then is δ - hyperbolic for a δ > 0 if and only if δ' - hyperbolic for a (possibly different ) δ ' > 0. In particular, a finite system of generators associated Cayley graph of a group is δ - hyperbolic for a δ > 0 if and only if this is true for any finite set of generators.

So you can then define independent of the chosen finite generating set S of a group G: the group G is hyperbolic if the Cayley graph is δ - hyperbolic for δ > 0.

Examples

  • Finite groups and cyclic groups are virtually hyperbolic, these groups are often referred to as a hyperbolic basic groups.
  • Finitely generated free groups are hyperbolic.
  • Fundamental groups of compact Riemannian manifolds of negative sectional curvature are hyperbolic. This includes in particular fundamental groups of compact hyperbolic manifolds, for example, fundamental groups of compact surfaces of negative Euler characteristic.
  • A " random " group is hyperbolic. That is precisely: For a ( arbitrary but fixed chosen ) natural number n and a d with 1/2
  • A group which contains as a subgroup, is not hyperbolic.

Applications

Various groups that can be formulated for arbitrary (and generally open ) conjectures were proved for the class of hyperbolic groups, using their special geometry. These include:

  • The Novikov conjecture
  • The tree - Connes conjecture
  • The Farrell -Jones Conjecture

Boundary at infinity

δ - hyperbolic spaces X have a mostly known as Gromov - boundary edge at infinity. This is defined as the set of equivalence classes of geodesic rays, where two rays are equivalent if and only if they have finite distance.

The choice of a fixed base point x in the X defining the topology of the following: The base of neighborhoods of a point p is used all the V ( p, r ) with, where V (P, R ), the set of q, such that p and q are represented by x outgoing geodesic rays is for. Here, the Gromov product. The topology is independent of the chosen x.

Quasi- isometric spaces have homeomorphic edges at infinity. In particular, the boundary of a hyperbolic group is well-defined (independent of the generating set S) as a boundary at infinity of the Cayley graph. Examples: for free groups is the boundary at infinity a Cantor set, for fundamental groups of compact n-dimensional Riemannian manifolds with negative sectional curvature of the boundary at infinity a n-1 - dimensional sphere, for " most" hyperbolic groups is the boundary at infinity a Menger sponge.

Quasi- isometries, in particular isometries of a δ - hyperbolic space X as homeomorphisms act on. In particular, every hyperbolic group G acts by isometries on its Cayley graph and so by homeomorphisms on the boundary at infinity. The effect of the hyperbolic group on the boundary at infinity is a " chaotic " dynamical system.

A hyperbolic group acts as a convergence group on its edge at infinity and this allows a topological characterization of hyperbolic groups: a group if and only hyperbolic if it acts as a uniform convergence group on a compact metrizable space.

Quasi-isometry
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