---
tag: BT8J7E5NVW
kind: statement
role: lemma
name: Greendlinger's lemma
sources:
  - wikipedia: en:19882883@r1368831820
    locator: Small cancellation theory, section Basic results of small cancellation theory > Greendlinger's lemma
    note: adopted
    mention: wikipedia://en/19882883#LVZQ
    sections: [Statement]
inventory: [CBH5EW0ABX]
---

## 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 − 3*λ*) ≥ 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 *D*<sub>1</sub> and *D*<sub>2</sub> in *D* such that for *j* = 1,2 the region *D*<sub>*j*</sub> intersects the boundary cycle ∂*D* of *D* in a simple arc whose length is bigger than (1 − 3*λ*)|∂*D*<sub>*j*</sub>|.

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.