The Whitehead asphericity conjecture claims that if (A. ∥ R) is an aspherical group presentation, then for every S ⊂ R the subpresentation (A ∥ S) is also aspherical. This conjecture is generalized for presentations of groups with periodic elements by introducing almost aspherical presentations (for example, every one-relator group is almost aspherical). 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. It is also proven that the generalized Whitehead asphericity conjecture holds for Ol'shanskii's presentations of free Burnside groups of large odd exponent, presentations of Tarski monsters and others.
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