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Group theory and generalizations Definition

Hyperbolic group

Builds on Metric spaceFinitely generated groupSubgroup generated by a subsetCayley graphQuasi-isometry

Definition

Let (X,d)(X, d) be a metric space that is geodesic: any two points are joined by a geodesic segment. A geodesic segment is an isometric image of a closed interval of ℝ\R. A geodesic triangle in XX consists of three points and three geodesic segments joining them in pairs. For δ≥0\delta \ge 0, a geodesic triangle is δ\delta-slim if each of its sides lies in the closed δ\delta-neighbourhood of the union of the other two sides. The space XX is δ\delta-hyperbolic if every geodesic triangle in XX is δ\delta-slim.

Let GG be a finitely generated group and let SS be a finite generating set. Give the Cayley graph Γ(G,S)\Gamma(G, S) the path metric in which each edge has length 11. The group GG is hyperbolic if Γ(G,S)\Gamma(G, S) is δ\delta-hyperbolic for some δ≥0\delta \ge 0.

Remarks

The definition does not depend on SS: Cayley graphs for different finite generating sets are quasi-isometric, and among geodesic spaces hyperbolicity is invariant under quasi-isometry, though δ\delta may change.

Example: the free group F2F_2 with a free basis SS. Its Cayley graph is a tree, in which every geodesic triangle is a tripod, so each side lies in the union of the other two and the graph is 00-hyperbolic.

Example: every finite group is hyperbolic, with δ\delta equal to the diameter of its Cayley graph.

Non-example: ℤ2\Z^2 is not hyperbolic. Its Cayley graph is quasi-isometric to the Euclidean plane. In the plane, take the triangle with vertices (0,0)(0, 0), (n,0)(n, 0) and (0,n)(0, n). The point (n/2,n/2)(n/2, n/2) of its hypotenuse is at distance n/2n/2 from the other two sides, which is unbounded in nn.

Equivalent definitions use the Gromov product, or ask that the group be finitely presented with a linear Dehn function. Such groups are also called word-hyperbolic or Gromov hyperbolic groups.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Cayley graph, concept, no evidence levelCayley graphconceptFinitely generated group, concept, no evidence levelFinitely generated gr…conceptMetric space, concept, no evidence levelMetric spaceconceptQuasi-isometry, concept, no evidence levelQuasi-isometryconceptSubgroup generated by a subset, concept, no evidence levelSubgroup generated by…conceptHyperbolic group, definition, no evidence levelHyperbolic groupdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Cayley graph concept
  2. Finitely generated group concept
  3. Metric space concept
  4. Quasi-isometry concept
  5. Subgroup generated by a subset concept
  6. Hyperbolic group definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Hyperbolic_group
About this page
TagE3ECET66DQ
URIm2://E3ECET66DQ
Kinddefinition
WritingW0 stub
MSC20F67, 20F65
DefinesHyperbolic group
UsesMetric space, Finitely generated group, Subgroup generated by a subset, Cayley graph, Quasi-isometry
Changed2 November 2026

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