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For the metric small cancellation condition $C^{\prime }(1/6)$, every nonempty freely reduced null-homotopic combinatorial word contains a segment of a cell boundary longer than half its perimeter. Such a segment can be replaced by the shorter complementary segment. The diagram explanation uses shells with at most three internal pieces: their internal length is less than half the perimeter. Degenerate diagrams require the disc diagram spur alternative.

The Greendlinger lemma says that a nonempty freely reduced null-homotopic combinatorial loop in a complex satisfying the metric small cancellation condition $C^{\prime }(1/6)$ contains a consecutive segment of a cell boundary longer than half that boundary. In a reduced disc diagram without boundary spurs this is supplied by a boundary disc diagram shell, or by the single-cell case. The long segment can be replaced by the strictly shorter complementary segment.

The useful precise diagram version also allows degenerate diagrams. A reduced disc diagram is either a single vertex, a single closed cell, a disc diagram ladder, or has at least three disc diagram spurs or shells whose inner paths consist of at most three pieces. This is the Greendlinger ladder theorem. For such a shell, writing the perimeter as $QP$, where $Q$ is the exterior arc and $P$ is the union of its interior pieces, the strict $C^{\prime }(1/6)$ inequalities giveA ladder is a chain of cells and possibly connecting edges, with the two boundary paths running along its opposite sides. In a nontrivial ladder with at least two blocks its two ends are shells or spurs. The single-cell case also has the long exterior arc conclusion when viewed against an appropriate boundary decomposition.

A tree diagram with no two-cells should not be described as containing a shell: it has a spur unless it is a point. Retaining the spur and ladder alternatives is essential for the quasiconvexity argument below.

Small cancellation theory controls overlap between relators or attaching paths of polygonal two-cells. Requiring overlaps to be small compared with cell perimeters gives strong disc diagram structure, shortening algorithms and geometric restrictions on subgroups. The metric small cancellation condition $C^{\prime }(1/6)$ is a central example.