COCKCROFT PRESENTATIONS

Let p be a prime or 0. A presentation P (defining a group G) is said t o be p-Cockcroft if the map pi(2)(P) --> H-2(P, Z(p)) is 0 (where Z(0) = Z). Moreover, if H is a subgroup of G then P is pH-Cockcroft if the covering of P (regarded as a 2-complex) corresponding to H is p-Cockc roft. These concepts, and other Cockcroft notions, are related to the minimality and efficiency of presentations, and to the ''relation gap' ' problem (in particular, a finite presentation is efficient if and on ly if it is p-Cockcroft for some prime p). If P is p-Cockcroft then th ere exist minimal subgroups H of G for which P is pH-Cockcroft, called p-Cockcroft thresholds (these were introduced by Harlander and Gilber t - Howie when p = 0). We investigate these thresholds. Finally, we ob tain necessary and sufficient conditions for the ''natural'' presentat ions for various group constructions ((generalized) graphs of groups, split extensions, direct products) to be p-Cockcroft. In particular, w e see why it is difficult to find minimal presentations for direct pro ducts.