Examples of Hyperbolic Groups
- Finite groups.
- Virtually cyclic groups.
- Finitely generated free groups, and more generally, groups that act on a locally finite tree with finite stabilizers.
- Most surface groups are hyperbolic, namely, the fundamental groups of surfaces with negative Euler characteristic. For example, the fundamental group of the sphere with two handles (the surface of genus two) is a hyperbolic group.
- Most triangle groups are hyperbolic, namely, those for which 1/l + 1/m + 1/n < 1, such as the (2,3,7) triangle group.
- The fundamental groups of compact Riemannian manifolds with strictly negative sectional curvature.
- Groups that act cocompactly 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.
- In some sense, "most" finitely presented groups with large defining relations are hyperbolic. See Random group.
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