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{{short description|Mathematical concept}}
{{Group theory sidebar |Discrete}}
In [[Group theory|group theory]], more precisely in [[Geometric group theory|geometric group theory]], a '''hyperbolic group''', also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated [[Group (mathematics)|group]] equipped with a [[Word metric|word metric]] satisfying certain properties abstracted from classical [[Hyperbolic geometry|hyperbolic geometry]]. The notion of a hyperbolic group was introduced and developed by {{harvs|first=Mikhail|last=Gromov|year=1987|txt=yes}}. The inspiration came from various existing mathematical theories: hyperbolic geometry but also low-dimensional topology (in particular the results of [[Biography:Max Dehn|Max Dehn]] concerning the [[Fundamental group|fundamental group]] of a hyperbolic [[Riemann surface]], and more complex phenomena in [[3-manifold|three-dimensional topology]]), and combinatorial group theory. In a very influential (over 1000 citations <ref>{{cite book |last1=Gromov |first1=Mikhail  |editor1-last=Gersten |editor1-first=S.M. |chapter=Hyperbolic Groups|title=Essays in Group Theory. Mathematical Sciences Research Institute Publications, vol 8. |date=1987 |publisher=Springer |location=New York, NY |pages=75–263 |ref=Gromov_1987 |url=https://www.ams.org/mathscinet-getitem?mr=919829}}</ref>) 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, [[Biography:William Thurston|William Thurston]], [[Biography:James W. Cannon|James W. Cannon]], [[Biography:Eliyahu Rips|Eliyahu Rips]], and many others.

== Definition ==

Let <math>G</math> be a finitely generated group, and <math>X</math> be its [[Cayley graph]] with respect to some finite set <math>S</math> of generators. The set <math>X</math> is endowed with its [[Distance (graph theory)|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 <math>G</math> is then said to be ''hyperbolic'' if <math>X</math> is a hyperbolic space in the sense of Gromov. Shortly, this means that there exists a <math>\delta > 0</math> such that any geodesic triangle in <math>X</math> is  <math>\delta</math>-thin, as illustrated in the figure on the right (the space is then said to be <math>\delta</math>-hyperbolic).

{{ Annotated image | caption=The δ-thin triangle condition
| image= Delta thin triangle condition.svg|
| width=230
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| annotations =
{{Annotation|105|12|<math>x</math>}}
{{Annotation|45|105|<math>y</math>}}
{{Annotation|205|110|<math>z</math>}}
{{Annotation|0|40|<math>B_\delta([x,y])</math>}}
{{Annotation|150|40|<math>B_\delta([z,x])</math>}}
{{Annotation|80|135|<math>B_\delta([y,z])</math>}}
}}

A priori this definition depends on the choice of a finite generating set <math>S</math>. That this is not the case follows from the two following facts:

*the Cayley graphs corresponding to two finite generating sets are always [[Quasi-isometry|quasi-isometric]] one to the other;
*any geodesic space which is quasi-isometric to a geodesic Gromov-hyperbolic space is itself Gromov-hyperbolic.

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

=== Remarks ===

The [[Švarc–Milnor lemma]]{{sfn|Bowditch|2006|loc=Theorem 3.6}} states that if a group <math>G</math> acts [[Group action (mathematics)#Remarkable properties of actions|properly discontinuously]] and with compact quotient (such an action is often called ''geometric'') on a proper length space <math>Y</math>, then it is finitely generated, and any Cayley graph for <math>G</math> is quasi-isometric to <math>Y</math>. Thus a group is (finitely generated and) hyperbolic if and only if it has a geometric action on a proper hyperbolic space.

If <math>G' \subset G</math> is a subgroup with finite index (i.e., the set <math>G/G'</math> is finite), then the inclusion induces a quasi-isometry on the vertices of any locally finite Cayley graph of <math>G'</math> into any locally finite Cayley graph of <math>G</math>. Thus <math>G'</math> is hyperbolic if and only if <math>G</math> itself is. More generally, if two groups are [[Commensurability (group theory)|commensurable]], then one is hyperbolic if and only if the other is.

== Examples ==

=== Elementary hyperbolic groups ===

The simplest examples of hyperbolic groups are finite groups (whose Cayley graphs are of finite diameter, hence <math>\delta</math>-hyperbolic with <math>\delta</math> equal to this diameter).

Another simple example is given by the infinite cyclic group <math>\Z</math>: the Cayley graph of <math>\Z</math> with respect to the generating set <math>\{ \pm 1 \}</math> is a line, so all triangles are line segments and the graph is <math>0</math>-hyperbolic. It follows that any group which is virtually cyclic (contains a copy of <math>\Z</math> of finite index) is also hyperbolic, for example the [[Infinite dihedral group|infinite dihedral group]].

Members in this class of groups are often called ''elementary hyperbolic groups'' (the terminology is adapted from that of actions on the hyperbolic plane).

=== Free groups and groups acting on trees ===

Let <math>S = \{a_1, \ldots, a_n\}</math> be a finite set and <math>F</math> be the [[Free group|free group]] with generating set <math>S</math>. Then the Cayley graph of <math>F</math> with respect to <math>S</math> is a locally finite [[Tree (graph theory)|tree]] and hence a 0-hyperbolic space. Thus <math>F</math> is a hyperbolic group.

More generally we see that any group <math>G</math> which acts properly discontinuously on a locally finite tree (in this context this means exactly that the stabilizers in <math>G</math> of the vertices are finite) is hyperbolic. Indeed, this follows from the fact that <math>G</math> has an invariant subtree on which it acts with compact quotient, and the Svarc—Milnor lemma. Such groups are in fact virtually free (i.e. contain a finitely generated free subgroup of finite index), which gives another proof of their hyperbolicity.

An interesting example is the [[Modular group|modular group]] <math>G = \mathrm{SL}_2(\mathbb Z)</math>: it acts on the tree given by the 1-skeleton of the associated [[Modular group#Tessellation of the hyperbolic plane|tessellation of the hyperbolic plane]] and it has a finite index free subgroup (on two generators) of index 6 (for example the set of matrices in <math>G</math> which reduce to the identity modulo 2 is such a group). Note an interesting feature of this example: it acts properly discontinuously on a hyperbolic space (the hyperbolic plane) but the action is not cocompact (and indeed <math>G</math> is ''not'' quasi-isometric to the hyperbolic plane).

=== Fuchsian groups ===

{{main article|Fuchsian group}}

Generalising the example of the modular group a [[Fuchsian group]] is a group admitting a properly discontinuous action on the hyperbolic plane (equivalently, a discrete subgroup of <math>\mathrm{SL}_2(\mathbb R)</math>). The hyperbolic plane is a <math>\delta</math>-hyperbolic space and hence the Svarc—Milnor lemma tells us that cocompact Fuchsian groups are hyperbolic.

Examples of such are the [[Fundamental group|fundamental group]]s of [[Surface (topology)#Closed surfaces|closed surfaces]] of negative [[Euler characteristic]]. Indeed, these surfaces can be obtained as quotients of the hyperbolic plane, as implied by the Poincaré—Koebe Uniformisation theorem.

Another family of examples of cocompact Fuchsian groups is given by [[Triangle group#The hyperbolic case|triangle groups]]: all but finitely many are hyperbolic.

=== Negative curvature ===

Generalising the example of closed surfaces, the fundamental groups of compact [[Riemannian manifold]]s with strictly negative [[Sectional curvature|sectional curvature]] are hyperbolic. For example, cocompact [[Lattice (discrete subgroup)|lattices]] in the orthogonal or [[Unitary group|unitary]] group preserving a form of signature <math>(n,1)</math> are hyperbolic.

A further generalisation is given by groups admitting a geometric action on a [[CAT(k) space]], when <math>k</math> is any negative number.<ref>for a proof that this includes the previous examples see https://lamington.wordpress.com/2012/10/17/upper-curvature-bounds-and-catk/</ref> There exist examples which are not commensurable to any of the previous constructions (for instance groups acting geometrically on hyperbolic [[Building (mathematics)|buildings]]).

=== Small cancellation groups ===

{{main article|Small cancellation theory}}

Groups having presentations which satisfy [[Small cancellation theory|small cancellation]] conditions are hyperbolic. This gives a source of examples which do not have a geometric origin as the ones given above. In fact one of the motivations for the initial development of hyperbolic groups was to give a more geometric interpretation of small cancellation.

=== Random groups ===

{{main article|Random group}}

In some sense, "most" finitely presented groups with large defining relations are hyperbolic. For a quantitative statement of what this means see [[Random group]].

=== Non-examples ===

*The simplest example of a group which is not hyperbolic is the [[Free abelian group|free rank 2 abelian group]] <math>\mathbb Z^2</math>. Indeed, it is quasi-isometric to the [[Euclidean plane]] which is easily seen not to be hyperbolic (for example because of the existence of [[Homothetic transformation|homotheties]]).
* More generally, any group which contains <math>\Z^2</math> as a [[Subgroup|subgroup]] is not hyperbolic.{{sfn|Ghys|de la Harpe|1990|loc=Ch. 8, Th. 37}}{{sfn|Bridson|Haefliger|1999|loc=Chapter 3.Γ, Corollary 3.10.}} In particular, [[Lattice (discrete subgroup)|lattices]] in higher rank semisimple Lie groups and the [[Fundamental group|fundamental group]]s <math>\pi_1(S^3\setminus K)</math> of nontrivial knot complements fall into this category and therefore are not hyperbolic. This is also the case for [[Mapping class group|mapping class group]]s of closed hyperbolic surfaces.
* The [[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) = <math>\Z^2</math>, this generalizes the previous example).
* A non-uniform lattice in a rank 1 simple Lie group is hyperbolic if and only if the group is isogenous to <math>\mathrm{SL}_2(\R)</math> (equivalently the associated symmetric space is the hyperbolic plane). An example of this is given by [[Hyperbolic link|hyperbolic]] [[Knot group|knot group]]s. Another is the [[Bianchi group]]s, for example <math>\mathrm{SL}_2(\sqrt{-1})</math>.

== Properties ==

=== Algebraic properties ===

*Hyperbolic groups satisfy the [[Tits alternative]]: they are either virtually solvable (this possibility is satisfied only by elementary hyperbolic groups) or they have a subgroup isomorphic to a nonabelian free group.
*Non-elementary hyperbolic groups are not [[Simple group|simple]] in a very strong sense: if <math>G</math> is non-elementary hyperbolic then there exists an infinite subgroup <math>H \triangleleft G</math> such that <math>H</math> and <math>G/H</math> are both infinite.
*It is not known whether there exists a hyperbolic group which is not [[Residually finite group|residually finite]].

=== Geometric properties ===

*Non-elementary (infinite and not virtually cyclic) hyperbolic groups have always exponential [[Growth rate (group theory)|growth rate]] (this is a consequence of the Tits alternative).
*Hyperbolic groups satisfy a linear [[Isoperimetric inequality|isoperimetric inequality]].{{sfn|Bowditch|2006|loc=(F4) in paragraph 6.11.2}}

=== Homological properties ===

*Hyperbolic groups are always finitely presented. In fact one can explicitly construct a complex (the Rips complex) which is [[Contractible space|contractible]] and on which the group acts geometrically{{sfn|Ghys|de la Harpe|1990|loc=Chapitre 4}} so it is of [[Finiteness properties of groups|type ''F''<sub>∞</sub>]]. When the group is torsion-free the action is free, showing that the group has finite [[Cohomological dimension|cohomological dimension]].
*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.{{sfn|Mineyev|2002}}

=== Algorithmic properties ===

*Hyperbolic groups have a solvable [[Word problem for groups|word problem]]. They are biautomatic and [[Automatic group|automatic]].{{sfn|Charney|1992}} 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.
*It was shown in 2010 that hyperbolic groups have a [[Philosophy:Decidability (logic)|decidable]] marked isomorphism problem.{{sfn|Dahmani|Guirardel|2011}} 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 [[Finite subdivision rule|subdivision rule]].{{sfn|Cannon|Swenson|1998}} This is related to [[Cannon's conjecture]].

== Generalisations ==

=== Relatively hyperbolic groups ===

{{main article|Relatively hyperbolic group}}

[[Relatively hyperbolic group]]s are a class generalising hyperbolic groups. ''Very'' roughly{{sfn|Bowditch|2012}} <math>G</math> is hyperbolic relative to a collection <math>\mathcal G</math> of subgroups if it admits a (''not necessarily cocompact'') properly discontinuous action on a proper hyperbolic space <math>X</math> which is "nice" on the boundary of <math>X</math> and such that the stabilisers in <math>G</math> of points on the boundary are subgroups in <math>\mathcal G</math>. This is interesting when both <math>X</math> and the action of <math>G</math> on <math>X</math> are not elementary (in particular <math>X</math> is infinite: for example every group is hyperbolic relatively to itself via its action on a single point!).

Interesting examples in this class include in particular non-uniform lattices in rank 1 semisimple [[Lie group]]s, for example fundamental groups of non-compact hyperbolic manifolds of finite volume. Non-examples are lattices in higher-rank Lie groups and mapping class groups.

=== Acylindrically hyperbolic groups ===

An even more general notion is that of an acylindically hyperbolic group.{{sfn|Osin|2016}} Acylindricity of an action of a group <math>G</math> on a metric space <math>X</math> is a weakening of proper discontinuity of the action.<ref>In some detail: it asks that for every <math>\varepsilon > 0</math> there exist <math>R, N > 0</math> such that for every two points <math>x, y \in X</math> which are at least <math>R</math> apart there are at most <math>N</math> elements <math>g \in G</math> satisfying <math>d(x, gx) < \varepsilon</math> and <math>d(y, gy) < \varepsilon </math>.</ref>

A group is said to be acylindrically hyperbolic if it admits a non-elementary acylindrical action on a (''not necessarily proper'') Gromov-hyperbolic space. This notion includes mapping class groups via their actions on [[Curve complex|curve complex]]es. Lattices in higher-rank Lie groups are (still!) not acylindrically hyperbolic.

=== CAT(0) groups ===

In another direction one can weaken the assumption about curvature in the examples above: a ''CAT(0) group'' is a group admitting a geometric action on a CAT(0) space. This includes [[Space group|Euclidean crystallographic groups]] and uniform lattices in higher-rank Lie groups.

It is not known whether there exists a hyperbolic group which is not CAT(0).<ref>{{cite web |title=Are all δ-hyperbolic groups CAT(0)? |date=February 10, 2015 |work=[[Company:Stack Exchange|Stack Exchange]] |url=https://math.stackexchange.com/q/1141526 }}</ref>

== Notes ==
<references/>

== References ==

* {{cite book
| last1 = Bridson
| first1 = Martin R.
| last2 = Haefliger
| first2 = André
| title = Metric spaces of non-positive curvature
| series = Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]
| volume = 319
| publisher = Springer-Verlag
| location = Berlin
| year = 1999
| isbn = 3-540-64324-9
| mr = 1744486
| doi = 10.1007/978-3-662-12494-9
}}

* {{cite book
| last = Bowditch
| first = Brian
| title = A course on geometric group theory
| series = MSJ Memoirs
| volume = 16
| publisher = Mathematical Society of Japan
| location= Tokyo
| year = 2006
| mr = 2243589
| isbn = 4-931469-35-3
| doi = 10.1142/e003
| url = http://www.warwick.ac.uk/~masgak/papers/bhb-ggtcourse.pdf
}}

* {{cite journal 
| first = Brian
| last = Bowditch 
| title = Relatively hyperbolic groups 
| journal = [[International Journal of Algebra and Computation]]
| year = 2012 
| volume = 22 
| issue = 3
| pages = 1250016, 66 pp 
| url = http://homepages.warwick.ac.uk/~masgak/papers/rhg.pdf
| doi=10.1142/S0218196712500166
| mr=2922380
| s2cid = 261118194 
}}

* {{cite journal | first1=James W. | last1=Cannon |  first2=Eric L. | last2=Swenson | title=Recognizing constant curvature discrete groups in dimension 3  | journal=[[Organization:Transactions of the American Mathematical Society|Transactions of the American Mathematical Society]] | volume=350 | year=1998 | issue=2 | pages=809–849 | mr=1458317 | doi=10.1090/S0002-9947-98-02107-2 | doi-access=free }}
* {{cite journal | 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 | mr=1157320 | issue=4 | s2cid=120654588 }}
* {{cite journal | last1=Dahmani | first1=François | last2=Guirardel | first2=Vincent | title=The isomorphism problem for all hyperbolic groups | journal=[[Geometric and Functional Analysis]] | arxiv=1002.2590 | year=2011 | volume=21 | issue=2 | pages=223–300 | doi=10.1007/s00039-011-0120-0 | s2cid=115165062 }}
* {{cite book | editor1-first=Étienne | editor1-last=Ghys  | editor1-link=Étienne Ghys| editor2-first=Pierre | editor2-last=de la Harpe | title=Sur les groupes hyperboliques d'après Mikhael Gromov | trans-title=Hyperbolic groups in the theory of Mikhael Gromov | language=fr | series=Progress in Mathematics | volume=83 | publisher=Birkhäuser Boston, Inc. | location=Boston, MA | year=1990 | isbn=0-8176-3508-4 | mr=1086648 | doi=10.1007/978-1-4684-9167-8 }}
* {{cite book |  first=Mikhail | last=Gromov | title=Essays in Group Theory | chapter=Hyperbolic Groups | pages=75–263 | series=Mathematical Sciences Research Institute Publications | volume=8 | publisher=Springer | location=New York | year=1987 | doi=10.1007/978-1-4613-9586-7_3 | mr=919829 | editor-last=Gersten | editor-first=Steve M. | isbn=0-387-96618-8 }}
* {{cite journal | first=Igor | last=Mineyev | title=Bounded cohomology characterizes hyperbolic groups | journal=[[Quarterly Journal of Mathematics]] | year=2002 | volume=53 | issue=1 | pages=59–73 | mr=1887670 | doi=10.1093/qjmath/53.1.59 }}
* {{cite journal 
| first = Denis
| last = Osin 
| title =  Acylindrically hyperbolic groups 
| journal = [[Organization:Transactions of the American Mathematical Society|Transactions of the American Mathematical Society]]
| year = 2016 
| volume = 368 
| issue = 2
| pages = 851–888
| arxiv = 1304.1246
| doi = 10.1090/tran/6343
| mr = 3430352
| s2cid = 21624534 
}}

== Further reading ==
* {{cite book | first1=Michel | last1=Coornaert | first2=Thomas | last2=Delzant | first3=Athanase | last3=Papadopoulos | title=Géométrie et théorie des groupes : les groupes hyperboliques de Gromov | language=fr | trans-title=Geometry and theory of groups: Gromov hyperbolic groups | series=Lecture Notes in Mathematics | volume=1441 | publisher=Springer-Verlag | location=Berlin | year=1990 | mr=1075994 | isbn=3-540-52977-2 | doi=10.1007/BFb0084913}}
* {{cite book | first1=Michel | last1=Coornaert | first2=Athanase | last2=Papadopoulos | title=Symbolic Dynamcis and Hyperbolic Groups | series=Lecture Notes in Mathematics | volume=1539 | publisher=Springer-Verlag | location=Berlin | year=1993 | isbn=3-540-56499-3 | mr=1222644 | doi=10.1007/BFb0092577}}
* {{springer|title=Gromov hyperbolic space|id=p/g110240}}

[[Category:Geometric group theory]]
[[Category:Metric geometry]]
[[Category:Properties of groups]]
[[Category:Hyperbolic geometry]]

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