Cockcroft presentations Original Research Article

Let p be a prime or 0. A presentation image (defining a group G) is said to be p-Cockcroft if the map π2(P) → H2(P, Zp) is 0 (where Z0 = Z). Moreover, if H is a subgroup of G then image is pH-Cockcroft if the covering of image (regarded as a 2-complex) corresponding to H is p-Cockcroft. 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 only if it is p-Cockcroft for some prime p). If image is p-Cockcroft then there exist minimal subgroups H of G for which image is pH-Cockcroft, called p-Cockcroft thresholds (these were introduced by Harlander and Gilbert — Howie when p = 0). We investigate these thresholds. Finally, we obtain necessary and sufficient conditions for the “natural” presentations for various group constructions ((generalized) graphs of groups, split extensions, direct products) to be p-Cockcroft. In particular, we see why it is difficult to find minimal presentations for direct products.