Small Cancellation Theory

Abstract

In 1911 M. Dehn posed the word and conjugacy problems for groups in general and provided algorithms which solved these problems for the fundamental groups of closed orientable two-dimensional manifolds. A crucial feature of these groups is that (with trivial exceptions) they are defined by a single relator r with the property that if s is a cyclic conjugate of r or r-1, with s ≠ r-1, there is very little cancellation in forming the product rs. Dehn’s algorithms have been extended to large classes of groups possessing presentations in which the defining relations have a similar small cancellation property. At first, investigations were concerned with the solution of the word problem for groups G presented as small cancellation quotients of a free group F. The theory was subsequently extended to the case where F is a free product, a free product with amalgamation, or an HNN extension. Moreover, strong results were obtained about algebraic properties; for example, one can classify torsion elements and commuting elements in small cancellation quotients.