# nLab combinatorial group theory

Combinatorial group theory is an approach to the theory of discrete groups based on presentations of groups, that is describing the groups by generators and relations. Despite the apparent simplicity of the idea at first look, some of the most basic problems of combinatorial group theory are difficult and unsolvable in general; for example S.P. Novikov and collaborators proved the existence of a group for which the word problem, that is the question of equality modulo the relations of elements given by generators, is algorithmically undecidable; in other words there is a presentation of this group such that no algorithm can take as an input two arbitrary words and decide in finite time if they represent the same element of the group.

While sometimes the presentation method is used in the theory of finite groups, most often not very deep facts are used; instead finite group theory is rather dominated with interaction with representation theory, ring theory, group cohomology and so on. Typical groups for which combinatorial methods give deep results are infinite discrete groups which are close in some sense to free groups, e.g. groups with a presentation having only one relation. Such groups often make their appearance as fundamental groups of interesting topological spaces. (Of course, all discrete groups can be realised as fundamental groups of spaces, so the exchange is two way.)

Nowadays combinatorial group theory uses fruitful connections to topology and graph theory: for example, the consequences if some group can be realized say as a fundamental group of a space in a nice way, or acts on a topological space are widely explored. They are apt to invite topological / geometric methods. In fact there is a hybrid field, geometric group theory (see also English Wikipedia).

Two of the classic texts on the subject are:

• W. Magnus, A. Karras, D. Solitar, Combinatorial group theory

• R. Lindon, P. Schupp, Combinatorial group theory, Springer 1977 (Russian transl. Mir, Moskva 1980)

For more elementary or introductory texts, see, for instance,

• D. L. Johnson, Presentations of Groups (London Mathematical Society Student Texts 15) 1990, Cambridge Univ. Press

• D. L. Johnson, Topics in Theory of Group Presentations, London Mathematical Society Lecture notes series 42) 1980, Cambridge Univ. Press