. This is interesting when both and the action of on are not elementary (in particular is infinite: for example every group is hyperbolic relatively to itself via its action on a single point!). Interesting examples in this class include in particular non-uniform lattices in rank 1 semisimple Lie groups, for example fundamental groups of non-compact hyperbolic manifolds of finite volume. Non-examples are lattices in higher-rank Lie groups and mapping class groups. Acylindrically hyperbolic groups An even more general notion is that of an acylindically hyperbolic group. Acylindricity of an action of a group on a metric space is a weakening of proper discontinuity of the action. A group is said to be acylindrically hyperbolic if it admits a non-elementary acylindrical action on a (not necessarily proper) Gromov-hyperbolic space. This notion includes mapping class groups via their actions on curve complexes. Lattices in higher-rank Lie groups are (still!) not acylindrically hyperbolic. CAT(0) groups In another direction one can weaken the assumption about curvature in the examples above: a CAT(0) group is a group admitting a geometric action on a CAT(0) space. This includes Euclidean crystallographic groups and uniform lattices in higher-rank Lie groups. It is not known whether there exists a hyperbolic group which is not CAT(0). Notes References Further reading Geometric group theory Metric geometry Properties of groups Combinatorics on words Hyperbolic metric space Hyperbolic geometry