Disc diagram

A disc diagram over a combinatorial 2-complex is a finite contractible planar combinatorial complex together with a combinatorial map to that complex. Its exterior boundary circuit records a null-homotopic path. Tree portions and cut vertices are allowed; the diagram need not be an embedded disk or map injectively. Its area is its number of two-cells.

A ladder is a disc diagram whose blocks form a linear chain, each block a two-cell or a connecting edge, with cells attached successively along interior arcs. Its boundary consists of two opposite side paths between its ends. Every cell meets both side paths. If cell perimeters are bounded by $L$, each point of one side is within $L/2$ of the other by traveling around a cell boundary; connecting edge paths coincide on the two sides.

A spur is an exposed degree-one vertex and its incident edge in a disc diagram. The exterior boundary runs out along that edge and immediately back. Thus a spur in the interior of a boundary side contradicts that side being a reduced combinatorial path or a metric geodesic. A diagram with no two-cells may be a tree and have spurs rather than shells.

A shell is an exposed two-cell in a disc diagram, with perimeter split as an exterior boundary arc $Q$ and an interior path $P$. An $i$-shell has $P$ consisting of $i$ interior pieces. Under $C^{\prime }(1/6)$, an $i$-shell with $i\le 3$ has $|P|<|\partial R|/2<|Q|$, so removal replaces a long exterior path by a strictly shorter path.

A closed combinatorial path in a 2-complex is null-homotopic exactly when it is the boundary path of a disc diagram over that complex. A finite cellular null-homotopy can be arranged into such a planar diagram; conversely the contractible diagram supplies a null-homotopy. In a group presentation, this is the geometric form of expressing a trivial word as a product of conjugates of relators.

A disc diagram is reduced if it has no adjacent pair of two-cells that can be folded together across their common boundary and removed while preserving the outside boundary. A minimum-area diagram is reduced. Among diagrams of minimum area, minimizing the number of edges removes unnecessary tree portions. Internal common arcs in a reduced diagram satisfy the applicable small cancellation theory piece bounds.