On aspherical presentations of groups

The Whitehead asphericity conjecture claims that ifA R is an aspherical group presentation, then for every S ⊂ R the subpresentationA S is also aspherical. This conjecture is generalized for presentations of groups with periodic elements by introduction of almost aspherical presen- tations. It is proven that the generalized Whitehead asphericity conjecture (which claims that every subpresentation of an almost aspherical presentation is also almost aspherical) is equivalent to the original Whitehead conjecture and holds for standard presentations of free Burnside groups of large odd exponent, Tarski monsters and some others. Next, it is proven that if theWhitehead conjecture is false, then there is an aspherical presentation E = A R∪z of the trivial group E, where the alphabet A is finite or countably infinite and z ϵ A, such that its subpresentationA R is not aspherical. It is also proven that if the Whitehead conjecture fails for finite presentations (i.e., with finite A and R), then there is a finite aspherical presentationA R, R = (R₁, R₂:, R_n), such that for every S⊆R the subpresentationA S i is aspherical and the subpresentationA R₁R₂, R₃,:, R_n of asphericalA R₁R₂; R₂;R₃;: R_n is not aspherical. Now suppose a group presenta- tion H =A R is aspherical, x ∉ A, W(A∪x) is a word in the alphabet (A∪x)^±1 with nonzero sum of exponents on x, and the group H naturally embeds in G =A∪x R∪W(A∪x). It is conjectured that the presentation G =A∪x R∪W(A∪x) is aspherical if and only if G is torsion free. It is proven that if this conjecture is false and G =A ∪x R∪W(A∪x) is a counterexample, then the integral group ring ℤ (G) of the torsion free group G will contain zero divisors. Some special cases where this conjecture holds are also indicated.