Towers, ladders and the B. B. Newman spelling theorem (Q2748391)

The spelling theorem of B. B. Newman states that in a group whose sole relator is of the form \(W^n\), any nontrivial word which represents the identity must contain a cyclic subword of \(W^{\pm n}\) longer than \(W^{n-1}\). It has been extended by various authors, notably by \textit{J. Howie} and \textit{S. J. Pride} [Invent. Math. 76, 55-74 (1984; Zbl 0544.20033)]. In the paper under review, these spelling theorems are reexamined using towers. Also, it is proven that a finitely generated group with one relator \(W^n\) is locally quasiconvex, provided that \(n\geq|W|\).NEWLINENEWLINENEWLINEA map \(A\to B\) of connected CW-complexes is a tower if it can be expressed as a composition of maps which are alternately inclusions of subcomplexes or infinite cyclic coverings. The authors review and slightly strengthen some methodology for towers due to Howie. The idea is to start with a disc (or spherical) diagram \(D\to X\), and go to a maximal tower \(Y\to X\) for which the diagram has a lift \(D\to Y\) to obtain information about the original diagram. This approach yields a version of the spelling theorem for staggered presentations, which are those for which the presentation \(2\)-complex admits linear orders of its \(2\)-cells and of a subset of its \(1\)-cells such that each \(2\)-cell has at least one of the ordered \(1\)-cells in its attaching map, and for any \(2\)-cells \(\alpha\) and \(\beta\), if \(\alpha<\beta\) then \((\min\alpha)<(\min\beta)\) and \((\max\alpha)<(\max\beta)\), where \((\min\alpha)\) and \((\max\alpha)\) are the least and greatest \(1\)-cells in the attaching map of \(\alpha\). The version states that if \(\langle x_1,\dots\mid R_1^{n_1},\dots\rangle\) is a staggered presentation, and \(U\) is a freely reduced cyclic word representing the identity, then \(U\) contains a subword \(S\) which is a subword of \(R_j^{\pm n_j}\) for some \(j\), with \(S\) longer than \(R_j^{n_j-1}\). The authors also give a proof of a similar result for pairs of words representing the same element of the group.NEWLINENEWLINENEWLINEIn the penultimate section of the paper, the authors discuss ladders. Roughly speaking, these are long, thin, disc diagrams which are never more than one relator wide. They develop strong restrictions on ladders associated to staggered presentations, leading to the local quasiconvexity result. The final section contains tower proofs of the Freiheitssatz and asphericity theorems for staggered presentations.