In the mathematical subject of group theory, small cancellation theory studies groups given by group presentations satisfying small cancellation conditions, that is where defining relations have "small overlaps" with each other. Small cancellation conditions imply algebraic, geometric and algorithmic properties of the group. Finitely presented groups satisfying sufficiently strong small cancellation conditions are word hyperbolic and have word problem solvable by Dehn's algorithm. Small cancellation methods are also used for constructing Tarski monsters, and for solutions of Burnside's problem.
History Some ideas underlying the small cancellation theory go back to the work of Max Dehn in the 1910s. Dehn proved that fundamental groups of closed orientable surfaces of genus at least two have word problem solvable by what is now called Dehn's algorithm. 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 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. The standard version of small cancellation theory, as it is used today, was developed by Martin Greendlinger in a series of papers in the early 1960s, who primarily dealt with the "metric" small cancellation conditions. 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, Schupp and Lyndon-Schupp, 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. Small cancellation theory was further generalized by Alexander Ol'shanskii who developed a "graded" version of the theory where the set of defining relations comes equipped with a filtration and where a defining relator of a particular grade is allowed to have a large overlap with a defining relator of a higher grade. Olshaskii used graded small cancellation theory to construct various "monster" groups, including the Tarski monster and also to give a new proof 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). Small cancellation theory supplied a basic set of examples and ideas for the theory of word-hyperbolic groups that was put forward by Gromov in a seminal 1987 monograph "Hyperbolic groups".
Main definitions The exposition below largely follows Ch. V of the book of Lyndon and Schupp.
Pieces Let
G = ⟨ X ∣ R ⟩ ( ∗ ) {\displaystyle G=\langle X\mid R\rangle \qquad (*)}
be a group presentation where R ⊆ F(X) is a set of freely reduced and cyclically reduced words in the free group F(X) such that R is symmetrized, that is, closed under taking cyclic permutations and inverses. 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. Note that if G = ⟨ X ∣ S ⟩ {\displaystyle G=\langle X\mid S\rangle } is a group presentation where the set of defining relators S is not symmetrized, we can always take the symmetrized closure R of S, where R consists of all cyclic permutations of elements of S and S−1. Then R is symmetrized and G = ⟨ X ∣ R ⟩ {\displaystyle G=\langle X\mid R\rangle } is also a presentation of G.
Metric small cancellation conditions Let 0 < λ < 1. 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|. Here |v| is the length of a word v. The condition C′(λ) is sometimes called a metric small cancellation condition.
Non-metric small cancellation conditions Let p ≥ 3 be an integer. A group presentation (∗) as above is said to satisfy the C(p) small cancellation condition if whenever r ∈ R and
r = u 1 … u m {\displaystyle r=u_{1}\dots u_{m}}
where ui are pieces and where the above product is freely reduced as written, then m ≥ p. That is, no defining relator can be written as a reduced product of fewer than p pieces. Let q ≥ 3 be an integer. A group presentation (∗) as above is said to satisfy the T(q) small cancellation condition if whenever 3 ≤ t < q and r1,...,rt in R are such that r1 ≠ r2−1,..., rt ≠ r1−1 then at least one of the products r1r2,...,rt−1rt, rtr1 is freely reduced as written. Geometrically, condition T(q) essentially means that if D is a reduced van Kampen diagram over (∗) then every interior vertex of D of degree at least three actually has degree at least q.
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