Combinatorial group theory

In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations. It is much used in geometric topology, the fundamental group of a simplicial complex having in a natural and geometric way such a presentation. A very closely related topic is geometric group theory, which today largely subsumes combinatorial group theory, using techniques from outside combinatorics besides.

It also comprises a number of algorithmically insoluble problems, most notably the word problem for groups; and the classical Burnside problem.