Relatively hyperbolic group

In mathematics, relatively hyperbolic groups form an important class of groups of interest for geometric group theory. The main purpose in their study is to extend the theory of Gromov-hyperbolic groups to groups ${\textstyle G}$ that may be regarded as hyperbolic assemblies of subgroups ${\textstyle H_{i}}$, called peripheral subgroups, in a way that enables "hyperbolic reduction" of problems for ${\textstyle G}$ to problems for the ${\textstyle H_{i}}$s.

Illustrative examples of relatively hyperbolic groups are provided by the fundamental groups of complete noncompact hyperbolic manifolds of finite volume. Further generalizations such as acylindrical hyperbolicity are also explored by current research.