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  "body": "\n## Statement\n\nThe 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\n\n**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\n\n$$\\left|v\\right| \u003e \\left(1-3\\lambda\\right)\\left|r\\right|$$\n\nNote 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.\n\nGreendlinger's lemma is obtained as a corollary of the following geometric statement:\n\nUnder 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*\u003csub\u003e1\u003c/sub\u003e and *D*\u003csub\u003e2\u003c/sub\u003e in *D* such that for *j* = 1,2 the region *D*\u003csub\u003e*j*\u003c/sub\u003e intersects the boundary cycle ∂*D* of *D* in a simple arc whose length is bigger than (1 − 3*λ*)|∂*D*\u003csub\u003e*j*\u003c/sub\u003e|.\n\nThis 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.\n",
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