CHAPTER 10 - FURTHER TOPICS

Summary

In this chapter we will consider a few other topics in combinatorial group theory where the methods have a topological flavour.

SMALL CANCELLATION THEORY

Small cancellation theory is one of the major aspects of combinatorial group theory. The methods are somewhat more geometrical than topological (insofar as it is possible to make such a distinction). I have something of a blind spot in this area, so I only summarise the results. For details see Lyndon and Schupp (1977). The paper by Greendlinger and Greendlinger (1984) simplifies one of the proofs given there.

Suppose that, in F(X), we have w u_ir_iu_i^-1, where w and each u_i and r_i are reduced. Then it is possible to make a diagram in the plane, composed of regions, edges, and vertices, with each edge being given a label from X∪X^-1, in such a way that the boundary of the whole diagram is a sequence of edges whose label (up to cyclic permutation) is w, while there are m regions, whose boundaries consist of sequences of edges whose labels are (up to cyclic permutation) the r_i.

Let R be a subset of F= F(X), and let N-<R>^F. We will assume that R is symmetrised; that is, that if r∈R then all cyclic permutations of r and of r^-1 are in R.

We define a piece (of R) to be an element u of F such that there are distinct r₁ and r₂ in R with r₁ - uv₁ and r₂ = uv₂ both reduced as written (r₂ is permitted to be r₁ or a cyclic permutation of r₁, or r₁^-1).