Combinatorial group theory

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.

Reading

The following are some of the basic texts in the subject.

1. J. Stillwell, Classical topology and combinatorial group theory, Graduate Texts in Mathematics 72, Springer 1980.
2. B. Chandler and W. Magnus, A history of combinatorial group theory, Springer 1982.
3. D. Johnson, Topics in the theory of group presentations, LMS Lecture Notes 42, CUP 1980.
4. R.C. Lyndon, Groups and geometry, LMS Lecture Notes 101, CUP 1985.
5. W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, Wiley 1966 (reprinted by Dover 1976).
6. J. Rotman, The theory of groups, an introduction, 3rd edition, Allyn and Bacon.
7. R.C. Lyndon and P.E. Schupp, Combinatorial group theory, Springer 1977.
8. J.-P. Serre, Trees, Springer 1980.
9. P. Cohen, Combinatorial group theory, a topological approach, LMS Student Texts 14, CUP 1989.
10. K.S. Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer 1982.
11. R. Crowell and R. Fox, Introduction to knot theory, Graduate Texts in Mathematics 57, Springer 1977.
12. E. Ghys and P. de la Harpe, Sur les Groupes Hyperboliques d'après Mikhael Gromov 83, Birkhauser 1990.
    

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