Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 133 / 2 / b / Solution

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.