Windmills and extreme 2-cells (Q716390)

Extreme \(2\)-cells in disk diagrams are, roughly speaking, those that are attached to the rest of the diagram along a small connected portion of their boundary cycle. A collection of extreme \(2\)-cells arranged around a disk conjures up the picture of a windmill and the authors make this notion precise in the context of disk diagrams: Let \(X\) be a \(2\)-complex, \(Y \hookrightarrow X\) be a subcomplex, \(Z \hookrightarrow X\) be its complement subcomplex, and let \(\Gamma = Y\cap Z\) be the subgraph of \(X\) which separates them. Let \(\phi:R\to X\) be a \(2\)-cell of \(X\). \(R\) is a windmill with respect to \(Z\) if \(\phi^{-1}(Z \setminus \Gamma)\) is connected and \(\phi^{-1}(\Gamma)\) is homeomorphic to a collection of isolated points in \(\partial R\) plus \(n\geq 2\) disjoint closed \(1\)-cells whose endpoints lie in \(\partial R\) and whose interiors lie entirely in the interior of \(R\). If each \(2\)-cell of \(X\) is a windmill with respect to \(Z\), then \(Z\) is a windmill in \(X\). The authors use windmills to prove that extreme \(2\)-cells exist. They also establish conditions on a \(2\)-complex \(X\) which imply that all minimal area disk diagrams over \(X\) with reduced boundary cycles have extreme \(2\)-cells. These lead to new results on coherence of various classes of one-relator groups, small cancellation groups, and groups with relatively staggered presentations.