Geometric group theory and combinatorial group theory are two closely related branches of mathematics, which study infinite discrete groups.
Geometric group theory uses topological and geometric methods to study groups; the main philosophy is to deduce information about a group by analyzing how it acts on topological spaces. Combinatorial group theory studies discrete groups as quotients of free groups, typically described using presentations. In the early 20th century, pioneering work of Dehn, Nielsen, Reidemeister and Schreier amongst others established a close correspondence between the two subjects. While some problems and methods are still discernably "more geometric" or "more combinatorial" than others, the fields are inextricably intertwined; they are now generally considered the same area of mathematics. Other closely related fields include algebraic topology, geometric topology and computational group theory.
Outline of topics to add:
What does it mean for a group to act on a space? What kinds of actions do we care about in geometric group theory?
Grouptheory can be considered the study of symmetry: the collection of symmetries of some object preserving some of its structure forms a group; in some sense all groups arise this way.
Geometric grouptheory and combinatorialgrouptheory are two closely related branches of mathematics, which study infinite discrete groups.
Geometric grouptheory uses topological and geometric methods to study groups; the main philosophy is to deduce information about a group by analyzing how it acts on topological spaces.
Combinatorialgrouptheory studies discrete groups as quotients of free groups, typically described using presentations.
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