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.
Read more about this topic: Hyperbolic Group
Famous quotes containing the words examples of, examples and/or groups:
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“Trees appeared in groups and singly, revolving coolly and blandly, displaying the latest fashions. The blue dampness of a ravine. A memory of love, disguised as a meadow. Wispy clouds—the greyhounds of heaven.”
—Vladimir Nabokov (1899–1977)