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. The inspiration came from various existing mathematical theories: hyperbolic geometry but also low-dimensional topology, and combinatorial group theory. In a very influential 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.
Definition
Let be a finitely generated group, and be its Cayley graph with respect to some finite set of generators. The set is endowed with its graph metric which turns it into a length space. The group is then said to be hyperbolic if is a hyperbolic space in the sense of Gromov. Shortly, this means that there exists a such that any geodesic triangle in is -thin, as illustrated in the figure on the right.A priori this definition depends on the choice of a finite generating set. That this is not the case follows from the two following facts:
- the Cayley graphs corresponding to two finite generating sets are always quasi-isometric one to the other;
- any geodesic space which is quasi-isometric to a geodesic Gromov-hyperbolic space is itself Gromov-hyperbolic.
Remarks
The Svarc--Milnor lemma states that if a group acting properly discontinuously and with compact quotient on a proper length space, then it is finitely generated, and any Cayley graph for is quasi-isometric to. Thus a group is hyperbolic if and only if it has a geometric action on a proper hyperbolic space.If is a subgroup with finite index, then the inclusion induces a quasi-isometry on the vertices of any Cayley graph of into any Cayley graph of. Thus is hyperbolic if and only if itself is. More generally if two groups are 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.Another very simple example is given by infinite cyclic group . It follows also that any group which is virtually cyclic is also hyperbolic, for example the infinite dihedral group.
Members in this class of groups are often called elementary hyperbolic groups.
Free groups and groups acting on trees
Let be a finite set and be the free group with generating set. Then the Cayley graph of with respect to is a locally finite tree and hence a 0-hyperbolic space. Thus is an hyperbolic group.More generally we see that any group which acts properly discontinuously on a locally finite tree is hyperbolic. Indeed, this follows from the fact that has an invariant subtree on which it acts with compact quotient, and the Svarc—Milnor lemma. Such groups are in fact virtually free, which gives another proof of their hyperbolicity.
An interesting example is the modular group : it acts on the tree given by the 1-skeleton of the associated tessellation of the hyperbolic plane and it has a finite index free subgroup of index 6. Note an interesting feature of this example: it acts properly discontinuously on a hyperbolic space but the action is not cocompact.
Fuchsian groups
Generalising the example of the modular group a Fuchsian group is a group admitting a properly discontinuous action on the hyperbolic plane. The hyperbolic plane is a -hyperbolic space and hence the Svarc—Milnor lemma tells us that cocompact Fuchsian groups are hyperbolic.Examples of such are the fundamental groups of 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 groups: all but finitely many are hyperbolic.
Negative curvature
Generalising the example of closed surfaces, the fundamental groups of compact Riemannian manifolds with strictly negative sectional curvature are hyperbolic. For example, cocompact lattices in the orthogonal or unitary group of a form of signature are hyperbolic.A further generalisation is given by groups admitting a geometric action on a CAT space. There exists examples which are not commensurable to any of the previous constructions.
Small cancellation groups
Groups having presentations which satisfy 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
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 rank 2 abelian group. Indeed, it is quasi-isometric to the Euclidean plane which is easily seen not to be hyperbolic.
- More generally, any group which contains as a subgroup is not hyperbolic. In particular, lattices in higher rank semisimple Lie groups and the fundamental groups of nontrivial knot complements fall into this category and therefore are not hyperbolic. This is also the case for mapping class groups of closed hyperbolic surfaces.
- The Baumslag–Solitar groups B and any group that contains a subgroup isomorphic to some B fail to be hyperbolic.
- A non-uniform lattice in a rank 1 simple Lie group is hyperbolic if and only if the group is isogenous to . An example of this is given by hyperbolic knot groups. Another is the Bianchi groups, for example.
Properties
Algebraic properties
- Hyperbolic groups satisfy the Tits alternative: they are either virtually solvable or they have a subgroup isomorphic to a nonabelian free group.
- Non-elementary hyperbolic groups are not simple in a very strong sense: if is non-elementary hyperbolic then there exists an infinite subgroup such that and are both infinite.
- It is not known whether there exists an hyperbolic group which is not residually finite.
Geometric properties
- Non-elementary hyperbolic groups have always exponential growth rate.
- Hyperbolic groups satisfy a linear isoperimetric inequality. In fact this is a characterisation of hyperbolicity.
Homological properties
- Hyperbolic groups are always finitely presented. In fact one can explicitly construct a complex which is contractible and on which the group acts geometrically so it is of type F∞. When the group is torsion-free the action is free, showing that the group has finite 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 ordinary cohomology is surjective in all degrees, or equivalently, in degree 2.
Algorithmic properties
- Hyperbolic groups have a solvable word problem. They are biautomatic and automatic. Indeed, they are 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 decidable marked isomorphism problem. It is notable that this means that the isomorphism problem, orbit problems and Whitehead's problem are all decidable.
- Cannon and Swenson have shown that hyperbolic groups with a 2-sphere at infinity have a natural subdivision rule. This is related to Cannon's conjecture.
Generalisations
Relatively hyperbolic groups
s are a class generalising hyperbolic groups. Very roughly is hyperbolic relative to a collection of subgroups if it admits a properly discontinuous action on a proper hyperbolic space which is "nice" on the boundary of and such that the stabilisers in of points on the boundary are subgroups in. This is interesting when both and the action of on are not elementary.Interesting examples in this class include in particular non-uniform lattices in rank 1 semisimple Lie groups, 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. Acylindricity of an action of a group on a metric space is a weakening of proper discontinuity of the action.A group is said to be acylindrically hyperbolic if it admits a non-elementary acylindrical action on a Gromov-hyperbolic space. This notion includes mapping class groups via their actions on curve complexes. Lattices in higher-rank Lie groups are not acylindrically hyperbolic.
CAT(0) groups
In another direction one can weaken the assumption about curvature in the examples above: a CAT group is a group admitting a geometric action on a CAT space. This includes 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.