Pieces

Let

${\displaystyle G=\langle X\mid R\rangle \qquad (*)}$

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 ${\displaystyle G=\langle X\mid S\rangle }$ 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 ${\displaystyle G=\langle X\mid R\rangle }$ 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

${\displaystyle r=u_{1}\dots u_{m}}$

where u_i 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,...,r_t in R are such that r₁ ≠ r₂⁻¹,..., r_t ≠ r₁⁻¹ then at least one of the products r₁r₂,...,r_t−1r_t, r_tr₁ 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.

Examples

• Let ${\displaystyle G=\langle a,b\mid aba^{-1}b^{-1}\rangle }$ be the standard presentation of the free abelian group of rank two. Then for the symmetrized closure of this presentation the only pieces are words of length 1. This symmetrized form satisfies the C(4)–T(4) small cancellation conditions and the C′(λ) condition for any 1 > λ > 1/4.
• Let ${\displaystyle G=\langle a_{1},b_{1},\dots ,a_{k},b_{k}\mid [a_{1},b_{1}]\cdot \dots \cdot [a_{k},b_{k}]\rangle }$, where k ≥ 2, be the standard presentation of the fundamental group of a closed orientable surface of genus k. Then for the symmetrization of this presentation the only pieces are words of length 1 and this symmetrization satisfies the C′(1/7) and C(8) small cancellation conditions.
• Let ${\displaystyle G=\langle a,b\mid abab^{2}ab^{3}\dots ab^{100}\rangle }$. Then, up to inversion, every piece for the symmetrized version of this presentation, has the form bⁱab^j or bⁱ, where 0 ≤ i,j ≤ 100. This symmetrization satisfies the C′(1/20) small cancellation condition.
• If a symmetrized presentation satisfies the C′(1/m) condition then it also satisfies the C(m) condition.
• Let r ∈ F(X) be a nontrivial cyclically reduced word which is not a proper power in F(X) and let n ≥ 2. Then the symmetrized closure of the presentation ${\displaystyle G=\langle X\mid r^{n}\rangle }$ satisfies the C(2n)^[16] and C′(1/n) small cancellation conditions.

Greendlinger's lemma

The main result regarding the metric small cancellation condition is the following statement (see Theorem 4.4 in Ch. V of ^[8]) which is usually called

Greendlinger's lemma: Let (∗) be a group presentation as above satisfying the C′(λ) small cancellation condition where 0 ≤ λ ≤ 1/6. Let w ∈ F(X) 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

${\displaystyle \left|v\right|>\left(1-3\lambda \right)\left|r\right|}$

Note that the assumption λ ≤ 1/6 implies that  (1 − 3λ) ≥ 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 D_j intersects the boundary cycle ∂D of D in a simple arc whose length is bigger than (1 − 3λ)|∂D_j|.

This result in turn is proved by considering a dual diagram for D. There one defines a combinatorial notion of curvature (which, by the small cancellation assumptions, is negative at every interior vertex), 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:

• Given a freely reduced word w on X^±1, construct a sequence of freely reduced words w = w₀, w₁, w₂,..., as follows.
• Suppose w_j is already constructed. If it is the empty word, terminate the algorithm. Otherwise check if w_j contains a subword v such that v is also a subword of some defining relator r = vu ∈ R such that |v| > |r|/2. If no, terminate the algorithm with output w_j. If yes, replace v by u⁻¹ in w_j, then freely reduce, denote the resulting freely reduced word by w_j+1 and go to the next step of the algorithm.

Note that we always have

|w₀| > |w₁| > |w₂| >...

which implies that the process must terminate in at most |w| steps. Moreover, all the words w_j 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(X) 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′(1/6) presentation Dehn's algorithm solves the word problem.

If a C′(1/6) presentation (∗) is finite (that is both X and R are finite), then Dehn's algorithm is an actual non-deterministic algorithm in the sense of recursion theory. However, even if (∗) is an infinite C′(1/6) 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′(1/6) or, more generally, C(6) presentation where every r ∈ R is not a proper power in F(X) 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 ${\displaystyle G=\langle X\mid S\rangle }$ is another presentation for G. Let Y be the presentation complex for this presentation. Then (see ^[17] and Theorem 13.3 in ^[9]), under the above assumptions on (∗), Y is a classifying space for G, that is G = π₁(Y) 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 (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. 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

• Let (∗) be a C′(1/6) presentation. Then an element g in G has order n > 1 if and only if there is a relator r in R of the form r = sⁿ in F(X) such that g is conjugate to s in G. In particular, if all elements of R are not proper powers in F(X) then G is torsion-free.
• If (∗) is a finite C′(1/6) presentation, the group G is word-hyperbolic.
• If R and S are finite symmetrized subsets of F(X) with equal normal closures in F(X) such that both presentations ${\displaystyle \langle X\mid R\rangle }$ and ${\displaystyle \langle X\mid S\rangle }$ satisfy the C′(1/6) condition then R = S.
• If a finite presentation (∗) satisfies one of C′(1/6), C′(1/4)–T(4), C(6), C(4)–T(4), C(3)–T(6) then the group G has solvable word problem and solvable conjugacy problem