In many situations groups arise by means of presentations. A presentation of a group G consists of a set of generators of G, together with a collection of relations amongst these generators such that any other relation amongst the generators is derivable (in a precise sense) from the given relations. As a simple example, the cyclic group of order n can be specified by a single generator a, and a single relation a raised to the nth power=1. Combinatorial group theory is the study of groups given by presentations. The term "combinatorial" derives from some of the techniques used which involve combining things together, rather than any (direct) link with the branch of mathematics called Combinatorics.
Combinatorial group theory has links with several other branches of mathematics.
Topology. Associated with any (connected) topological space X is its fundamental group. This can often be specified by means of a presentation. Algebraic information about this group can be used to obtain topological information about X.
Logic. One connection comes through a consideration of algorithmic problems (or decision problems) in group theory. These problems are concerned with determining whether or not there are algorithms (i.e. computer programs) which can be used for obtaining group-theoretic information from a presentation. Another connection comes through a famous theorem of G. Higman which links the subgroup structure of groups given by certain types of presentations with recursive function theory. A third, more recent, connection is with language theory through the study of "automatic groups".
Other branches of group theory. One can use combinatorial group theory to construct groups with pre-assigned properties. Also, combinatorial group theory has close links with the cohomology theory of groups.
Number Theory. Certain groups which arise in number theory can be studied using combinatorial group theory.
Many techniques of combinatorial group theory are purely algebraic, and it is possible to achieve much using these. However, in recent years many techniques involving geometric ideas have emerged, and are proving more and more fruitful. These geometric techniques involve graph theory, the theory of tesselations of various surfaces, and covering space theory, to name a few.
The following are some of the basic texts in the subject.