Greendlinger's lemma

Statement

The main result regarding the metric small cancellation condition is the following statement (see Theorem 4.4 in Ch. V of Roger C. Lyndon and Paul Schupp, Combinatorial group theory. Reprint of the 1977 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. ) 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

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

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.