Small cancellation theory

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.

A CW subcomplex has no missing shells if a cell with a shell-length boundary arc contained in the CW subcomplex is itself contained in the CW subcomplex. For $C^{\prime }(1/6)$ it suffices that every excluded cell have a complementary boundary arc of length at least half its perimeter with no edge in the CW subcomplex: an arc lying in the CW subcomplex then has length at most half and cannot be a shell's exterior arc.

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.

A reduced $C^{\prime }(1/6)$ disc diagram is a single vertex, a single closed cell, a disc diagram ladder, or contains at least three exposed disc diagram spurs and/or disc diagram shells with at most three interior pieces. This is the strong diagram form of the Greendlinger lemma. It follows from the combinatorial curvature inequality after suppressing valence-two vertices: if there are not three positively curved exposed features, all remaining blocks must occur in a chain. In a nontrivial multi-block ladder its two ends are exposed features. Marking two boundary corners and excluding shells/spurs in the side interiors therefore forces the ladder alternative. This is the width-one case of McCammond and Wise, Theorem 9.4; reduced $C^{\prime }(1/6)$ diagrams satisfy its arc-reduced $C(6)$–$T(3)$ hypotheses.

The condition $C^{\prime }(\lambda )$ requires that every piece in small cancellation theory occurring in a cell boundary have length strictly less than $\lambda$ times that cell's perimeter. Length counts edges. For $\lambda =1/6$, an inner path made of at most three pieces has length strictly less than half the perimeter, so the complementary outer path is strictly longer. The attaching paths are combinatorial immersions and the piece convention is part of the chosen presentation or polygonal complex.

A piece is a nontrivial combinatorial path that occurs in two essentially distinct positions on oriented cell attaching paths. These may be different cells or distinct admissible occurrences on one attaching path; occurrences identified by the defining attaching-path symmetry are not counted twice. In a symmetrized group presentation, it is a common initial segment of distinct relator words. Internal arcs of a reduced disc diagram are pieces, because otherwise the neighboring cells form a cancellable pair.