for is quasi-isometric to . Thus a group is (finitely generated and) hyperbolic if and only if it has a geometric action on a proper hyperbolic space. If is a subgroup with finite index (i.e., the set is finite), then the inclusion induces a quasi-isometry on the vertices of any locally finite Cayley graph of into any locally finite Cayley graph of . Thus is hyperbolic if and only if itself is. More generally, if two groups are commensurable, then one is hyperbolic if and only if the other is. Examples Elementary hyperbolic groups The simplest examples of hyperbolic groups are finite groups (whose Cayley graphs are of finite diameter, hence -hyperbolic with equal to this diameter). Another simple example is given by the infinite cyclic group : the Cayley graph of with respect to the generating set is a line, so all triangles are line segments and the graph is -hyperbolic. It follows that any group which is virtually cyclic (contains a copy of of finite index) is also hyperbolic, for example the infinite dihedral group. Members in this class of groups are often called elementary hyperbolic groups (the terminology is adapted from that of actions on the hyperbolic plane). Free groups and groups acting on trees Let be a finite set and be the free group with generating set . Then the Cayley graph of with respect to is a locally finite tree and hence a 0-hyperbolic space. Thus is a hyperbolic group. More generally we see that any group which acts properly discontinuously on a locally finite tree (in this context this means exactly that the stabilizers in of the vertices are finite) is hyperbolic. Indeed, this follows from the fact that has an invariant subtree on which it acts with compact quotient, and the Svarc—Milnor lemma. Such groups are in fact virtually free (i.e. contain a finitely generated free subgroup of finite index), which gives another proof of their hyperbolicity. An interesting example is the modular group : it acts on the tree given by the 1-skeleton of the associated tessellation of the hyperbolic plane and it has a finite index free subgroup (on two generators) of index 6 (for example the set of matrices in which reduce to the identity modulo 2 is such a group). Note an interesting feature of this example: it acts properly discontinuously on a hyperbolic space (the hyperbolic plane) but