Geometric group theory

Geometric group theory and combinatorial group theory are two closely related branches of mathematics, which study infinite discrete groups.

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?