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?
• The Cayley graph as the canonical space to act on. The adjacency matrix of a Cayley graph allows number-theoretic methods to be applied as well, via spectral graph theory.
• The Ping-Pong lemma , which is the main way to exhibit a group as a free product
• Finiteness properties
• Amenability, as it is studied by geometric group theory

Geometric group theory is mainly the study of some particular examples:

• The infinite cyclic group Z
• Free groups
• Free products
• Out(F_n) (via Outer space)
• Hyperbolic groups
• Mapping class groups (automorphisms of surfaces)
• Braid groups
• General Artin groups
• Thompson's group F
• CAT(0) groups
• Soluble groups?
• Arithmetic groups?
• (Bi)automatic groups?

External links

• The Geometric Group Theory Page
• What is Geometric Group Theory?
• Open Problems in combinatorial and geometric group theory