the action is not cocompact (and indeed is not quasi-isometric to the hyperbolic plane). Fuchsian groups Generalising the example of the modular group a Fuchsian group is a group admitting a properly discontinuous action on the hyperbolic plane (equivalently, a discrete subgroup of ). The hyperbolic plane is a -hyperbolic space and hence the Svarc—Milnor lemma tells us that cocompact Fuchsian groups are hyperbolic. Examples of such are the fundamental groups of closed surfaces of negative Euler characteristic. Indeed, these surfaces can be obtained as quotients of the hyperbolic plane, as implied by the Poincaré—Koebe Uniformisation theorem. Another family of examples of cocompact Fuchsian groups is given by triangle groups: all but finitely many are hyperbolic. Negative curvature Generalising the example of closed surfaces, the fundamental groups of compact Riemannian manifolds with strictly negative sectional curvature are hyperbolic. For example, cocompact lattices in the orthogonal or unitary group of a form of signature are hyperbolic. A further generalisation is given by groups admitting a geometric action on a CAT(k) space. There exist examples which are not commensurable to any of the previous constructions (for instance groups acting geometrically on hyperbolic buildings). Small cancellation groups Groups having presentations which satisfy small cancellation conditions are hyperbolic. This gives a source of examples which do not have a geometric origin as the ones given above. In fact one of the motivations for the initial development of hyperbolic groups was to give a more geometric interpretation of small cancellation. Random groups In some sense, "most" finitely presented groups with large defining relations are hyperbolic. For a quantitative statement of what this means see Random group. Non-examples The simplest example of a group which is not hyperbolic is the free rank 2 abelian group . Indeed, it is quasi-isometric to the Euclidean plane which is easily seen not to be hyperbolic (for example because of the existence of homotheties). More generally, any group which contains as a subgroup is not hyperbolic. In particular, lattices in higher rank semisimple Lie groups and the fundamental groups of nontrivial knot complements fall into this category and therefore are not hyperbolic. This is also the case for mapping class groups of closed hyperbolic surfaces. The Baumslag–Solitar groups B(m,n) and any group that contains a subgroup isomorphic to some B(m,n) fail to be hyperbolic (since B(1,1) = , this generalizes the previous example). A non-uniform lattice in a rank 1