Hyperbolic group - Encyclopedia of Mathematics.

In continuation of (JOH04, OH07), we prove that existentially closed CSA-groups have the independence property. This is done by showing that there exist words having the independence property relative to the class of torsion-free hyperbolic groups.

In this paper, we generalize the classical definition of Gromov hyperbolicity to the context of directed graphs and we extend one of the main results of the theory: the equivalence of the Gromov.

Hyperbolic group explained. In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry.The notion of a hyperbolic group was introduced and developed by.

Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces).

In this paper, we detail the geometrical approach of small cancellation theory used by Delzant and Gromov to provide a new proof of the infiniteness of free Burnside groups and periodic quotients of torsion-free hyperbolic groups.

Similar Schottky-type arguments are widely used in geometric group theory, particularly for subgroups of word-hyperbolic groups and for automorphism groups of trees. Ping-pong lemma is also used for studying Schottky-type subgroups of mapping class groups of Riemann surfaces, where the set on which the mapping class group acts is the Thurston.

Abstract: Suppose G is a Gromov hyperbolic group, and the boundary at infinity of G is quasisymmetrically homeomorphic to an Ahlfors Q-regular metric 2-sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and isometrically on hyperbolic 3-space.

The study of word hyperbolic groups is a prominent topic in geometric group theory; however word hyperbolic groups are defined by a geometric condition which does not extend naturally to.

In mathematics, the concept of a relatively hyperbolic group is an important generalization of the geometric group theory concept of a hyperbolic group.The motivating examples of relatively hyperbolic groups are the fundamental groups of complete noncompact hyperbolic manifolds of finite volume.

In this paper we study the hyperbolicity in the Gromov sense of Riemann surfaces. We deduce the hyperbolicity of a surface from the hyperbolicity of its “building block components”. We also prove the equivalence between the hyperbolicity of a Riemann surface and the hyperbolicity of some graph associated with it.

Let H be a properly discontinuous group of isometries of a negatively curved (Gromov hyperbolic) metric space X.We give equivalent conditions on H to be quasi-convex. The main application of this is to give alternate definitions of quasi-convex, or rational subgroups of negatively curved (word hyperbolic) groups.

James W. Cannon (born January 30, 1943) is an American mathematician working in the areas of low-dimensional topology and geometric group theory.He was an Orson Pratt Professor of Mathematics at Brigham Young University.