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:
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“Only the groups which exclude us have magic.”
—Mason Cooley (b. 1927)