Small cancellation conditions imply algebraic, geometric and algorithmic properties of the group.
Some ideas underlying the small cancellation theory go back to the work of Max Dehn in the 1910s.
His proof involved drawing the Cayley graph of such a group in the hyperbolic plane and performing curvature estimates via the Gauss–Bonnet theorem for a closed loop in the Cayley graph to conclude that such a loop must contain a large portion (more than a half) of a defining relation.
A 1949 paper of Tartakovskii[2] was an immediate precursor for small cancellation theory: this paper provided a solution of the word problem for a class of groups satisfying a complicated set of combinatorial conditions, where small cancellation type assumptions played a key role.
In particular, Greendlinger proved that finitely presented groups satisfying the C′(1/6) small cancellation condition have word problem solvable by Dehn's algorithm.
The theory was further refined and formalized in the subsequent work of Lyndon,[6] Schupp[7] and Lyndon-Schupp,[8] who also treated the case of non-metric small cancellation conditions and developed a version of small cancellation theory for amalgamated free products and HNN-extensions.
Olshaskii used graded small cancellation theory to construct various "monster" groups, including the Tarski monster[10] and also to give a new proof[11] that free Burnside groups of large odd exponent are infinite (this result was originally proved by Adian and Novikov in 1968 using more combinatorial methods).
A nontrivial freely reduced word u in F(X) is called a piece with respect to (∗) if there exist two distinct elements r1, r2 in R that have u as maximal common initial segment.
Presentation (∗) as above is said to satisfy the C′(λ) small cancellation condition if whenever u is a piece with respect to (∗) and u is a subword of some r ∈ R, then |u| < λ|r|.
The main result regarding the metric small cancellation condition is the following statement (see Theorem 4.4 in Ch.
For any symmetrized group presentation (∗), the following abstract procedure is called Dehn's algorithm: Note that we always have which implies that the process must terminate in at most |w| steps.
More generally, it is possible to define various sorts of local "curvature" on any van Kampen diagram to be - very roughly - the average excess of vertices + faces − edges (which, by Euler's formula, must total 2) and, by showing, in a particular group, that this is always non-positive (or – even better – negative) internally, show that the curvature must all be on or near the boundary and thereby try to obtain a solution of the word problem.