The asphericity and freiheitssatz for certain lot-presentations of groups

Let U be a word in letters x₁^±1, . . . , x_m^±1, m > 2, and a group G be given by presentation G = 〈x, . . . , x_m||Ux_iU^-1 = X_i+1, i = 1, . . . , m -1〉. It is proven that this presentation is aspherical provided the word U does not have the form U₂U₁, where U₁ is a word in letters x₁^±1, . . . , x_m-1^±1 and U₂ is a word in letters x₂^±1, . . . , _m^±1. It is also proven that the (images of) x₁, . . . , x_m-1 freely generate a free subgroup of G if and only if the word U does not have the foregoing form U₂U₁.