Gromov hyperbolic group

E911231

geometric group theory concept group theory concept mathematical concept

A Gromov hyperbolic group is a finitely generated group whose Cayley graph exhibits negative curvature–like properties, leading to rich geometric, dynamical, and algorithmic behavior.

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Observed surface forms (1)

Surface form	Occurrences
Gromov hyperbolic groups	1

Statements (84)

Predicate	Object
instanceOf	geometric group theory concept ⓘ group theory concept ⓘ mathematical concept ⓘ
appearsInWork	Gromov's paper "Hyperbolic groups" in Essays in Group Theory (1987) NERFINISHED ⓘ
characterizedBy	Cayley graph is Gromov hyperbolic for one (equivalently any) finite generating set ⓘ existence of δ such that all geodesic triangles in Cayley graph are δ-thin ⓘ
field	dynamical systems ⓘ geometric group theory ⓘ geometry ⓘ group theory ⓘ theoretical computer science ⓘ
generalizationOf	fundamental groups of closed negatively curved Riemannian manifolds ⓘ
hasDefinition	a finitely generated group whose Cayley graph is Gromov hyperbolic ⓘ a finitely generated group whose Cayley graph is a δ-hyperbolic metric space for some δ ≥ 0 ⓘ
hasExample	cocompact lattice in a rank-one simple Lie group ⓘ free group of rank at least 2 ⓘ fundamental group of a closed hyperbolic n-manifold ⓘ fundamental group of a closed hyperbolic surface ⓘ fundamental groups of many graphs of groups with hyperbolic vertex groups and quasi-convex edge groups ⓘ small cancellation groups satisfying C′(1/6) ⓘ word-hyperbolic Coxeter groups ⓘ
hasProperty	Bowditch boundary coincides with Gromov boundary ⓘ Cannon–Thurston maps exist in many natural extensions ⓘ Cayley graph has thin triangles ⓘ Cayley graph is a geodesic metric space ⓘ Cayley graph is roughly isometric to a proper hyperbolic metric space ⓘ Cayley graph is δ-hyperbolic ⓘ Cayley graph satisfies Rips condition ⓘ Cayley graph satisfies slim triangle condition ⓘ Morse geodesics ⓘ Poisson boundary often identifies with Gromov boundary NERFINISHED ⓘ acts as a convergence group on its boundary ⓘ automatic group ⓘ biautomatic group ⓘ boundary at infinity is compact ⓘ boundary at infinity is locally connected in many cases ⓘ boundary at infinity is metrizable ⓘ boundary at infinity is perfect if group is non-elementary ⓘ boundary dynamics are topologically mixing for many actions ⓘ cohomological dimension equals topological dimension of boundary plus one (under mild hypotheses) ⓘ every abelian subgroup is virtually cyclic ⓘ every amenable hyperbolic group is virtually cyclic ⓘ exponential growth (if infinite and non-virtually cyclic) ⓘ finite asymptotic dimension ⓘ finite intersection of quasi-convex subgroups is quasi-convex ⓘ finite number of conjugacy classes of finite subgroups ⓘ finite presentation ⓘ finitely generated ⓘ growth series is rational for many hyperbolic groups ⓘ is relatively hyperbolic only to finite subgroups ⓘ linear Dehn function ⓘ linear isoperimetric inequality ⓘ negatively curved in the large-scale sense ⓘ no subgroups isomorphic to ℤ² unless virtually cyclic ⓘ non-elementary hyperbolic groups are not amenable ⓘ non-elementary hyperbolic groups have exponential growth of conjugacy classes ⓘ non-elementary hyperbolic groups have uncountably many pairwise non-conjugate subgroups ⓘ quasi-convex subgroups are finitely generated ⓘ quasi-convex subgroups are hyperbolic ⓘ quasi-isometry invariant ⓘ random walks have positive drift ⓘ satisfies Tits alternative ⓘ satisfies linear isodiametric inequality ⓘ satisfies linear-time solution to word problem with respect to automatic structure ⓘ satisfies property of stability of quasi-geodesics ⓘ solvable conjugacy problem ⓘ solvable word problem ⓘ strongly geodesically automatic ⓘ virtually torsion-free ⓘ visual metrics can be defined on the boundary ⓘ word-hyperbolic ⓘ
introducedBy	Mikhail Gromov NERFINISHED ⓘ
introducedIn	1980s ⓘ
namedAfter	Mikhail Gromov NERFINISHED ⓘ
relatedConcept	Bowditch boundary NERFINISHED ⓘ CAT(-1) group NERFINISHED ⓘ Cannon conjecture NERFINISHED ⓘ Gromov boundary NERFINISHED ⓘ Gromov hyperbolic space NERFINISHED ⓘ automatic group ⓘ convergence group ⓘ quasi-convex subgroup ⓘ relatively hyperbolic group ⓘ small cancellation theory ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Dehn function → connectedTo → Gromov hyperbolic group ⓘ

Mikhail Gromov → notableFor → Gromov hyperbolic group ⓘ

this entity surface form: Gromov hyperbolic groups