Chapter VII - Small cancellation groups

Summary

The Moving Finger writes; and, having writ, Moves on: nor all thy Piety nor Wit Shall lure it back to cancel half a Line, Nor all thy Tears wash out a Word of it.

(Fitzgerald: The Rubá iyát of Omar Khayyám)

Given a group G = <X|R>, suppose that the relators in R are all cyclically reduced and that R is symmetrized, that is, if r belongs to R, then so do all cyclic conjugates of r and r^-1. Then G satisfies a small cancellation hypothesis if the amount of cancelling in forming any product rs (r,s ∈ R, r ≠ s^-1 in F(X)) is limited in one of various senses to be made precise in §24. The formulation of these hypotheses is inspired by the properties of the planar diagrams studied in §23. The latter boast a degree of intrinsic usefulness, and may be thought of as portions of the Cayley diagram of G adapted to fit inside R² The power of the hypotheses derives from Euler's formula for planar graphs, which explains the innocent but pervasive topological overtones encountered in this branch of combinatorial group theory.

The conclusions that may be inferred from small cancellation hypotheses form an interrelated hierarchy of properties such as:

1. G is infinite,

2. the torsion elements in G can be classified,

3. G is SQ-universal,’ that is, any countable group can be embedded in some factor group of G,

4. G contains a non-cyclic free subgroup, and

5. G has soluble word problem, that is, there is an algorithm for deciding whether or not any given word in F(X) belongs to R.