Small cancellation theory


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 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' 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.
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
be a group presentation where R ⊆ F is a set of freely reduced and cyclically reduced words in the free group F such that R is symmetrized, that is, closed under taking cyclic permutations and inverses.
A nontrivial freely reduced word u in F 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 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 is also a presentation of G.

Metric small cancellation conditions

Let 0 < λ < 1. Presentation as above is said to satisfy the C' 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' 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 small cancellation condition if whenever r ∈ R and
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 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 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.

Examples

Greendlinger's lemma

The main result regarding the metric small cancellation condition is the following statement which is usually called
Greendlinger's lemma:
Let be a group presentation as above satisfying the C' small cancellation condition where 0 ≤ λ ≤ 1/6. Let w ∈ F be a nontrivial freely reduced word such that w = 1 in G. Then there is a subword v of w and a defining relator r ∈ R such that v is also a subword of r and such that
Note that the assumption λ ≤ 1/6 implies that ≥ 1/2, so that w contains a subword more than a half of some defining relator.
Greendlinger's lemma is obtained as a corollary of the following geometric statement:
Under the assumptions of Greendlinger's lemma, let D be a reduced van Kampen diagram over with a cyclically reduced boundary label such that D contains at least two regions. Then there exist two distinct regions D1 and D2 in D such that for j = 1,2 the region Dj intersects the boundary cycle ∂D of D in a simple arc whose length is bigger than |∂Dj|.
This result in turn is proved by considering a dual diagram for D. There one defines a combinatorial notion of curvature, and one then obtains a combinatorial version of the Gauss–Bonnet theorem. Greendlinger's lemma is proved as a consequence of this analysis and in this way the proof evokes the ideas of the original proof of Dehn for the case of surface groups.

Dehn's algorithm

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. Moreover, all the words wj represent the same element of G as does w and hence if the process terminates with the empty word, then w represents the identity element of G.
One says that for a symmetrized presentation Dehn's algorithm solves the word problem in G if the converse is also true, that is if for any freely reduced word w in F this word represents the identity element of G if and only if Dehn's algorithm, starting from w, terminates in the empty word.
Greendlinger's lemma implies that for a C' presentation Dehn's algorithm solves the word problem.
If a C' presentation is finite, then Dehn's algorithm is an actual non-deterministic algorithm in the sense of recursion theory. However, even if is an infinite C' presentation, Dehn's algorithm, understood as an abstract procedure, still correctly decides whether or not a word in the generators X±1 represents the identity element of G.

Asphericity

Let be a C' or, more generally, C presentation where every r ∈ R is not a proper power in F then G is aspherical in the following sense. Consider a minimal subset S of R such that the symmetrized closure of S is equal to R. Thus if r and s are distinct elements of S then r is not a cyclic permutation of s±1 and is another presentation for G. Let Y be the presentation complex for this presentation. Then, under the above assumptions on, Y is a classifying space for G, that is G = π1 and the universal cover of Y is contractible. In particular, this implies that G is torsion-free and has cohomological dimension two.

More general curvature

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 and, by showing, in a particular group, that this is always non-positive internally, show that the curvature must all be on or near the boundary and thereby try to obtain a solution of the word problem. Furthermore, one can restrict attention to diagrams that do not contain any of a set of "regions" such that there is a "smaller" region with the same boundary.

Other basic properties of small cancellation groups

Examples of applications of small cancellation theory include: