Geometric Group Theory
Geometric group theory studies groups by connecting their algebraic properties to the topological and geometric properties of spaces on which they act. Sometimes the group itself is treated as a geometric object; occasionally auxiliary structures on the group, such as orders, arise naturally. The field emerged as a distinct area in the late 1980s and has many interactions with other parts of mathematics, including computational group theory, low-dimensional topology, algebraic topology, hyperbolic geometry, the study of Lie groups and their discrete subgroups and K-theory.