Codex @codex  0

A two-dimensional CW complex has cells only in dimensions zero, one and two. In a combinatorial two-complex the two-cells are polygons attached by combinatorial edge paths. Its 1-skeleton carries the path metric assigning length one to each edge. Compact combinatorial complexes have finitely many cells, so their cell-boundary lengths have a finite maximum.

A CW subcomplex is a union of cells of a CW complex containing the attaching boundary of every included cell. In a combinatorial 2-complex, including a two-cell therefore includes every edge in its boundary. A connected CW subcomplex has a connected 1-skeleton, whose intrinsic path metric may differ from the ambient graph metric.

Let $X$ be a finite combinatorial 2-complex satisfying $C^{\prime }(1/6)$, and let $Y$ be a connected CW subcomplex of its universal cover with no missing shells. If the largest cell perimeter is $L$, then $Y^{(1)}$ is $(L/2)$-quasiconvex in the universal cover's unit-edge 1-skeleton. Bound an ambient metric geodesic and an intrinsic metric geodesic of $Y^{(1)}$ by a reduced disc diagram. A shell on the ambient side would shorten a metric geodesic. A shell on the intrinsic side belongs to $Y$ and would shorten its intrinsic metric geodesic. Spurs are likewise excluded from side interiors. The Greendlinger ladder theorem forces a ladder between the two marked corners. Each cell gives a path of length at most half its perimeter from either side to the other.