Cockcroft presentations
Let \(p\) be a prime or 0. A presentation \(\mathcal P\) is said to be \(p\)-Cockcroft if the map \(\pi_2({\mathcal P})\to H_2({\mathcal P}, Z_p)\) is 0. This concept is related to the minimality and efficiency of presentations and to the relation gap problem. In fact, a finite presentation is efficient iff it is \(p\)-Cockcroft for some prime \(p\). Thus the \(p\)-Cockcroft notion is a natural extension of the efficiency notion to arbitrary (not necessarily finite) presentations. If \(\mathcal P\) is \(p\)-Cockcroft then there exist minimal subgroups in the group being presented for which the corresponding covering complex is \(p\)-Cockcroft (called \(p\)-Cockcroft thresholds). The authors investigate the \(p\)-Cockcroft property and thresholds in terms of combinatorial homotopy (pictures). In particular they show that \(p\)-Cockcroft thresholds are non-unique. Finally, they obtain necessary and sufficient conditions for the presentations of various group constructions (e.g. generalized graph of groups) to be \(p\)-Cockcroft.
scientific article; zbMATH DE number 858647
scientific article