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Hyperbolic group

The meaning of «hyperbolic group»

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 Mikhail Gromov (1987). The inspiration came from various existing mathematical theories: hyperbolic geometry but also low-dimensional topology (in particular the results of Max Dehn concerning the fundamental group of a hyperbolic Riemann surface, and more complex phenomena in three-dimensional topology), and combinatorial group theory. In a very influential (over 1000 citations [1]) chapter 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.

Let G {\displaystyle G} be a finitely generated group, and X {\displaystyle X} be its Cayley graph with respect to some finite set S {\displaystyle S} of generators. The set X {\displaystyle X} is endowed with its graph metric (in which edges are of length one and the distance between two vertices is the minimal number of edges in a path connecting them) which turns it into a length space. The group G {\displaystyle G} is then said to be hyperbolic if X {\displaystyle X} is a hyperbolic space in the sense of Gromov. Shortly, this means that there exists a δ > 0 {\displaystyle \delta >0} such that any geodesic triangle in X {\displaystyle X} is δ {\displaystyle \delta } -thin, as illustrated in the figure on the right (the space is then said to be δ {\displaystyle \delta } -hyperbolic).

A priori this definition depends on the choice of a finite generating set S {\displaystyle S} . That this is not the case follows from the two following facts:

Thus we can legitimately speak of a finitely generated group G {\displaystyle G} being hyperbolic without referring to a generating set. On the other hand, a space which is quasi-isometric to a δ {\displaystyle \delta } -hyperbolic space is itself δ ′ {\displaystyle \delta '} -hyperbolic for some δ ′ > 0 {\displaystyle \delta '>0} but the latter depends on both the original δ {\displaystyle \delta } and on the quasi-isometry, thus it does not make sense to speak of G {\displaystyle G} being δ {\displaystyle \delta } -hyperbolic.

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