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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 (mathematics)|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 (mathematician)|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.
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| == Definitions ==
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| 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.
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| 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 [[isometry|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''.
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| {{ Annotated image | caption=The δ-slim triangle condition
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| | image= Delta thin triangle condition.svg|
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| | width=230
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| | height = 155
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| | image-width = 200
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| | image-left = 30
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| | annotations =
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| {{Annotation|105|12|<math>x</math>}}
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| {{Annotation|45|105|<math>y</math>}}
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| {{Annotation|205|110|<math>z</math>}}
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| {{Annotation|0|40|<math>B_\delta([x,y])</math>}}
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| {{Annotation|150|40|<math>B_\delta([z,x])</math>}}
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| {{Annotation|80|135|<math>B_\delta([y,z])</math>}}
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| }}
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| The first approach to hyperbolicity is based on the ''slim triangles'' condition and is generally credited to Rips. Let δ > 0 be fixed. A geodesic triangle is '''δ-slim''' if each side is contained in a <math>\delta</math>-neighborhood of the other two sides:
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| :::<math>[x,y] \subseteq B_{\delta}([y,z]\cup[z,x]),</math>
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| :::<math>[y,z]\subseteq B_{\delta}([z,x]\cup[x,y]),</math>
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| :::<math>[z,x]\subseteq B_{\delta}([x,y]\cup[y,z]). </math>
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| 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-isometry|quasi-isometries]]. Therefore, the property of being a hyperbolic group depends only on the group itself.
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| === Remark ===
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| By imposing the slim triangles condition on geodesic metric spaces in general, one arrives at the more general notion of [[Δ-hyperbolic space|<math>\delta</math>-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.
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| == Examples of hyperbolic groups ==
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| * [[Finite group]]s.
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| * [[Virtually cyclic group]]s.
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| * [[Finitely generated group|Finitely generated]] [[free group]]s, and more generally, groups that [[group action|act]] on a locally finite [[tree (graph theory)|tree]] with finite stabilizers.
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| * Most ''[[surface]] groups'' are hyperbolic, namely, the [[fundamental group]]s of surfaces with negative [[Euler characteristic]]. For example, the fundamental group of the sphere with two handles (the surface of [[genus (topology)|genus]] two) is a hyperbolic group.
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| * Most [[triangle group]]s <math>\Delta(l,m,n)</math> are hyperbolic, namely, those for which 1/''l'' + 1/''m'' + 1/''n'' < 1, such as the [[(2,3,7) triangle group]].
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| * The fundamental groups of compact [[Riemannian manifold]]s with strictly negative [[sectional curvature]].
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| * Groups that act [[cocompact]]ly 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.
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| * In some sense, "most" finitely presented groups with large defining relations are hyperbolic. See [[Random group]].
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| == Examples of non-hyperbolic groups ==
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| * The [[free abelian group|free rank 2 abelian group]] '''Z'''<sup>2</sup> is not hyperbolic.
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| * More generally, any group which contains '''Z'''<sup>2</sup> as a [[subgroup]] is not hyperbolic.<ref>Ghys and de la Harpe, Ch. 8, Th. 37; Bridson and [[André Haefliger|Haefliger]], Chapter 3.Γ, Corollary 3.10.</ref> In particular, [[lattice (discrete subgroup)|lattices]] in higher rank [[semisimple Lie group]]s and the [[fundamental group]]s ''π''<sub>1</sub>(''S''<sup>3</sup>−''K'') of nontrivial [[knot (mathematics)|knot]] complements fall into this category and therefore are not hyperbolic.
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| * [[Baumslag–Solitar group]]s ''B''(''m'',''n'') and any group that contains a subgroup isomorphic to some ''B''(''m'',''n'') fail to be hyperbolic (since ''B''(1,1) = '''Z'''<sup>2</sup>, this generalizes the previous example).
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| * A non-uniform lattice in rank 1 semisimple Lie groups is hyperbolic if and only if the associated symmetric space is the hyperbolic plane.
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| == Homological characterization ==
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| In 2002, I. Mineyev showed that hyperbolic groups are exactly those finitely generated groups for which the comparison map between the [[bounded cohomology]] and [[Group cohomology|ordinary cohomology]] is surjective in all degrees, or equivalently, in degree 2.
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| == Properties ==
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| Hyperbolic groups have a solvable [[word problem for groups|word problem]]. They are [[biautomatic group|biautomatic]] and [[automatic group|automatic]].:<ref name=charney>{{citation | last=Charney | first=Ruth | title=Artin groups of finite type are biautomatic | journal=Mathematische Annalen | volume= 292 | year=1992 | doi=10.1007/BF01444642 | pages=671–683}}</ref> indeed, they are [[automatic group|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.
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| In a 2010 paper,<ref>Dahmani, F.; Guirardel, V. - On the Isomorphism Problem in all Hyperbolic Groups, arXiV: [http://arxiv.org/abs/1002.2590 1002.2590]</ref> it was shown 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.
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| Cannon and Swenson have shown <ref name="CS">J. W. Cannon and E. L. Swenson, ''Recognizing constant curvature discrete groups in dimension 3''. [[Transactions of the American Mathematical Society]] 350 (1998), no. 2, pp. 809–849.</ref> that hyperbolic groups with a 2-sphere at infinity have a natural [[finite subdivision rule|subdivision rule]]. This is related to [[James W. Cannon|Cannon's Conjecture]].
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| == Generalizations ==
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| {{main|Relatively hyperbolic group}}
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| 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 knot]]s, which are not hyperbolic in the sense of Gromov.
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| A group ''G'' is '''relatively hyperbolic''' with respect to a subgroup ''H'' if, after contracting the Cayley graph of ''G'' along ''H''-[[coset]]s, 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.
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| == Notes ==
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| <references/>
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| == References ==
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| * [[Mikhail Gromov (mathematician)|Mikhail Gromov]], ''Hyperbolic groups.'' Essays in group theory, 75–263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987.
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| * {{cite book
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| | last = Bridson
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| | first = Martin R.
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| | coauthors = Haefliger, André
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| | title = Metric spaces of non-positive curvature
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| | series = Grundlehren der Mathematischen Wissenschaften 319
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| | publisher = Springer-Verlag
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| | location = Berlin
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| | year = 1999
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| | isbn = 3-540-64324-9
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| | nopp = true
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| | page = xxii+643
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| }} {{MathSciNet|id=1744486}}
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| * Igor Mineyev, ''Bounded cohomology characterizes hyperbolic groups.'', Quart. J. Math. Oxford Ser., 53(2002), 59-73.
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| == Further reading ==
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| * 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
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| * Michel Coornaert, Thomas Delzant and Athanase Papadopoulos, "Géométrie et théorie des groupes : les groupes hyperboliques de Gromov", Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990, x+165 pp. MR 92f:57003, ISBN 3-540-52977-2
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| * Michel Coornaert and Athanase Papadopoulos, Symbolic dynamics and hyperbolic groups. Lecture Notes in Mathematics. 1539. Springer-Verlag, Berlin, 1993, viii+138 pp. ISBN 3-540-56499-3
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| * {{springer|title=Gromov hyperbolic space|id=p/g110240}}
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| [[Category:Geometric group theory]]
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| [[Category:Metric geometry]]
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| [[Category:Properties of groups]]
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| [[Category:Combinatorics on words]]
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